Բազմաչափ կոմպլեքս անալիզի հիմունքները

EREVANI PETAKAN HAMALSARAN

A. I. PETROSYAN

BAZMA A KOMPLEQS

ANALIZI HIMUNQNER

Hastatva  HH kruyan gituyan naxararuyan komic orpes dasagirq buheri fizikamaematikakan masnagituyunneri usanoneri hamar

EREVANI HAMALSARANI HRATARAK UYUN

EREVAN { 2007

HTD 517 GMD 22.16 P 505

Hratarakuyan  eraxavorel EPH maematikayi fakulteti xorhurd

Xmbagir` fiz.-ma. git. ekna u, docent G. V. Miqayelyan Graxosner` fiz.-ma. git. doktor, profesor H. M. Hayrapetyan fiz.-ma. git. doktor, profesor A. H. Hovhannisyan

Petrosyan A. I.

P 505 Bazmaa kompleqs analizi himunqner: { Er.: Er ani hamals. hrat., 2007, 194 j: Girq a ajinn  hayeren lezvov bazmaa kompleqs analizic: Ayn bakaca  ors glxic: A ajin glux nvirrva  mi qani o oxakani holomorf funkcianeri parzaguyn hatkuyunnerin karo  gtakar linel na o maematikakan masnagitacum staco usanoneri hamar: Erkrord errord gluxnerum aradrva en Cn -um holomorfuyan ti{ ruyneri nranc hamareq holomorf u ucik ps dou ucik tiruyneri hatkuyunner: orrord glux nvirva  integralayin nerkayacumne{ rin: Girq parunakum  bazmaiv xndirner, oronc lu um knpasti ha{ mapatasxan emaneri yuracman: Naxatesva  hamalsaranneri maematikayi, kira akan maema{ tikayi fizikayi fakultetneri, inpes na harakic masnagituyun{ neri usanoneri, aspirantneri gitaxatoneri hamar:

P

1602070000 704(02)07

ISBN 978 · 5-8084-0862-3

GMD 22.16

c ⃝

A. Petrosyan, 2007 .

Bovandakuyun Naxaban

Glux 1. Holomorf funkcianer

§ 1. § 2. § 3. § 4. § 5. § 6. § 7. § 8. § 9.

Kompleqs tara uyun . . . . . . . . . . . . . . . . . . . Astianayin arqer . . . . . . . . . . . . . . . . . . . . Holomorf funkciayi sahmanum . . . . . . . . . . . . . Plyuriharmonik funkcianer . . . . . . . . . . . . . . . Holomorf funkciayi zroner . . . . . . . . . . . . . . . Koii bana  nra parzaguyn kira uyunner . Ayl arqer . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorf artapatkerumner . . . . . . . . . . . . . . . Gaa ar meromorf funkciayi masin . . . . . . . . . Xndirner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Glux 11. Holomorfuyan tiruyner § 10. § 11. § 12. § 13.

Analitik arunakuyun . Holomorfuyan tiruyner Holomorf u ucikuyun . . Ow ucikuyun st L ii . Xndirner . . . . . . . . . . . . . .

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Bovandakuyun

Glux 111. Ps dou ucik tiruyner § 14. § 15. § 16. § 17. § 18. § 19.

Subharmonik funkcianer . . . . Plyurisubharmonik funkcianer Ow ucik funkcianer . . . . . . . Ps dou ucik tiruyner . . . . . Ow ucik tiruyner . . . . . . . . . Holomorfuyan aan . . . . . . Xndirner . . . . . . . . . . . . . . . . . .

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Glux 1Մ. Integralayin nerkayacumner § 20. Diferencial  . . . . . . . § 21. Koi{Puankarei eorem § 22. Martineli{Boxneri bana § 23. Lerei bana  . . . . . . . . § 24. Veyli bana  . . . . . . . . § 25. ungei tiruyner . . . . . § 26. ∂¯ -xndir . . . . . . . . . . . § 27. Ke nfunkcia . . . . . . . .

 . Xndirner . . . . . . . . . . . . . . . . .

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Grakanuyun

A arkayakan cank

Naxaban Bazmaa kompleqs analiz (mi qani kompleqs o oxakani analitik funkcianeri tesuyun), i tarberuyun miaa

depqi, maematikayi hamematabar nor bnagava : naya or ayd tesuyan a ajin hetazotuyunner er an en ekel de Vayertrasi Puankarei axatuyunnerum, nra akan zargacum sksvel  miayn ancyal dari 60-akan vakan{ nerin: Inpes haytni , mi qani o oxakani holomorf funkcianeri hatkuyunneric ater unen irenc nmanak mek o oxakani depqum: Da bacatrvum  nranov, or mi qani o oxakani depqum Koi- imani paymanner kazmum en diferencial hava{ sarumneri gerorova hamakarg: Aynpes or, bazmaa kompleqs analizi haskacuyunnern u xndirner himnakanum en handi{ sanum miaa depqi uaki ndhanracum, ayl unen yuraha{ tuk drva q, or mek o oxakani hamar kam animast , kam parzunak: Mek o oxakani analitik funkcianeri tesuyun Ha{ yastanum bavakanin zargaca  haytni  ir hetazotu{ yunnerov ardyunqnerov: Bazmaa kompleqs analiz, linelov maematikayi hamematabar nor bnagava , a avel zargana{ lu mitum uni: Husov em, or suyn girq kunena ir nanakuyun ayd gor um: Ays dasnac heinak tariner arunak dasavandel  Er ani petakan hamalsarani maematikayi fakulteti usanoneri orakavorman barracman fakulteti unkndir{ neri hamar: Nra bovandakuyun hamapatasxanum  mae{ matikayi fakultetum dasavandvo <bazmaa kompleqs ana{

Naxaban

liz> a arkayi ragrin: Heinak ir xorin norhakaluyunn  haytnum fiz.-ma. git. ekna u, docent M. A. Mkrtyanin suyn grqi veraberyal di{ touyunneri, gtakar xorhurdneri hamar, fiz.-ma. git. ekna u, docent G. V. Miqayelyanin girq xmbagrelu hamar:

GLUX

HOLOMORF FUNKCIANER

§

1. Kompleqs tara uyun

1. Cn tara uyun: Cn -ov nanakvum  C kompleqs har{

uyan n -patik dekartyan artadryal` Cn = C · · × C} : | × ·{z n

Cn -i keter n kompleqs veri kargavorva n -yakner en` z = = (z1 , z2 , . . . , zn ) : Ayd tara uyunum bnakan  ov nermu vum 

g ayin ka ucva q kompleqs veri dati vra, masnavorapes, C1 = C : Cn - kareli  nuynacnel R2n -i het, or bakaca  x = (x1 , x2 , . . . , x2n ) keteric ori vra nermu va  kompleqs ka ucva q, aysinqn trva ` zk = xk + ixn+k , k = 1, . . . , n : gtagor elu enq na xn+k = yk nanakum, aynpes or zk = = xk + iyk : 2. Haruyunner Cn -um: Kompleqs ka ucva qi nermu u{

m Cn tara uyunum a aj  berum anhamaa uyun, rinak, x1 xn+1 koordinatner kazmum en kompleqs vi irakan ke

maser, isk x1 -n u xn -` o: Dra het anqov Cn tara uyan haruyunneri mej xaxtvum  havasarazoruyun: Ditarkenq 2m -a ani haruyun` } { 2n ∑ S = x: αik xk = βi , i = 1, . . . , 2n − 2m , k=1

(1.1)

§ 1.

Kompleqs tara uyun

orte αik βi -er irakan ver en rang ∥αik ∥ = 2n − 2m : (1.1)-i aj masum masnakco havasarumner grenq kompleqs ko{ ordinatneri tesqov, teadrelov nranc mej xk = (zk + z̄k ) xn+k

= (zk − z̄k ) : Kstananq` 2i { } n ∑ ′ S = z: aik zk + aik z̄k = bi , i = 1, . . . , n − m ,

(1.2)

k=1

orte

aik ,

kompleqs ver en: S a h m a n u m 1.1. S haruyun kovum  kompleqs haruyun, ee (1.2)-i mej bacakayum en z̄k o oxakanner, aysinqn` a′ik = 0 . m iv hamarvum  nra kompleqs a oa{ kanuyun: Henc dranq en Cn tara uyan <iskakan> haruyunner: O kompleqs haruyan rinakner karo en a ayel irakan Rnx (ke Rny ) n a ani haruyunner, oronq

va en irakan (hamapatasxanabar` ke ) a ancqneri vra: Kompleqs miaa haruyun anvanum en na kompleqs ui, isk (n − 1) -a ani haruyun` kompleqs hiperhar{ uyun: z 0 ketov ancno kompleqs ui kareli  tal ereq tarber  erov. 1. havasarumneri hamakargov` a′ik

bi -er

{ } n ∑ S = z: aik (zk − zk ) = 0, i = 1, . . . , n − 1 , k=1

rang ∥aik ∥ = n − 1,

2. parametrakan tesqov`

{ } S = z : zk = zk0 + ak ζ, ζ ∈ C, k = 1, . . . , n ,

Glux Լ. Holomorf funkcianer

3. kanonakan havasarumnerov` {

S=

z1 − z10 zn − zn0 z: = ··· = a1 an

} :

V a r  u  y u n 1.1. Cuyc tal, or Cn tara uyan kama{

yakan erku ketov ancnum  miak kompleqs ui:

3. Metrika: Cn -um rmityan skalyar artadryal  ko{

vum

⟨z, w⟩ =

n ∑

(1.3)

zk w̄k ,

k=1

or aknhaytoren bavararum  het yal paymannerin` ⟨w, z⟩ = ⟨z, w⟩,

⟨λz, w⟩ = λ⟨z, w⟩

kamayakan

λ ∈ C vi hamar: Dicuq zk = xk + ixn+k wk = uk + iun+k : (1.3)-ic stanum enq ⟨z, w⟩ =

2n ∑

xk uk + i

k=1

2n ∑

xn+k uk − xk un+k ,

k=1

orteic er um , or rmityan skalyar artadryali Re ⟨z, w⟩ ira{ kan mas z w vektorneri vklidyan skalyar artadryaln  R2n tara uyan mej: rmityan skalyar artadryalov a ajaca

|z|2 = ⟨z, z⟩ =

n ∑

|zk |2 =

2n ∑

x2k

k=1

k=1

norm handisanum  z vektori erkaruyun hamapatasxan metrikan`

R2n -um:

v u n u∑ |zk − wk |2 ρ(z, w) = |z − w| = t k=1

Owremn,

§ 1.

Kompleqs tara uyun

hamnknum  R2n -um sovorakan vklidyan metrikayi het: Cn tara uyunum erbemn ditarkum en na δ(z, w) = max |zk − wk | 16k6n

metrikan, or kovum  polidiskayin metrika: 4. Parzaguyn tiruyner: Ditarkenq parzaguyn tiruy{

ner 1.

Cn -um:

G u n d  ( a ∈ Cn kentronov

r

a avov)

B(a, r) = {z ∈ Cn : |z − a| < r}

bazmuyunn : Da sovorakan 2n a ani gund  R2n tara u{ yan mej: Nra S(a, r) = {z ∈ Cn : |z − a| = r} ezr 2n − 1 a ani gndolort : 2. P o l i d i s k  ( a ∈ Cn kentronov r = (r1 , . . . , rn ) bazmaa avov) U (a, r) = {z ∈ Cn : |zj − aj | < rj , j = 1, . . . , n}

bazmuyunn : Da haruyan vra aj kentronov rj a avov sovorakan rjanneri dekartyan artadryal : Polidiski ∂U ezr bnakan  ov trohvum  Γk = {z : |zk − ak | = rk , |zj − aj | 6 rj , j ̸= k}

bazmuyunneri, oroncic yuraqanyur 2n − 1 a ani  ∂U = n n ∪ ∩ = Γk : Ayd bazmuyunneri Γ = Γk ndhanur mas, ori k=1 k=1 a oakanuyun n , kovum  polidiski henq ( îñòîâ ) : n > 1 depqum henq kazmum  tiruyi ezri mi oqr mas, sakayn at harcerum na katarum akan der:

Glux Լ. Holomorf funkcianer

3. B a z m a g l a n kovum  har tiruyneri dekartyan artadryal, aysinqn` D = D1 × · · · × Dn ,

Dj ⊂ C, j = 1, . . . , n :

Masnavorapes, ee Dk -er rjanner en, stanum enq polidisk: 4. e y n h a r t i t i r u y  n e r (kam n -rjana tiruyner), oronc kentron a ∈ Cn ketn . dranq ayn tiruynern en, oronq amen mi z 0 = (z10 , . . . , zn0 ) keti het mekte parunakum en na ( ) z = a1 + (z10 − a1 )eiθ1 , . . . , an + (zn0 − an )eiθn , 0 6 θj 6 2π

tesqi bolor keter: Aysinqn, { } z 0 ∈ D ⇒ z : |zk − ak | = |zk0 − ak |, k = 1, . . . , n ⊂ D : D -n

kovum  lriv eynharti tiruy, ee { } z 0 ∈ D ⇒ z : |zk − ak | 6 |zk0 − ak |, k = 1, . . . , n ⊂ D :

Ee a = 0 , apa stacvum en 0 kentronov eynharti ti{ ruyner. dranq amen mi z = (z1 , . . . , zn ) keti het mekte pa{ runakum en bolor ayn keter, oronc hamar |zk | -er nuynn en, isk argumentner kamayakan en, aysinqn, aydpisi tiruyne{ r liovin orovum en irenc patkano keteri modulnerov: Ayd pata ov nranc usumnasiruyun kareli  katarel Rn ta{ ra uyan Rn+ = R × · · · × R+ drakan ktantum, katarelov | + {z } n

artapatkerum: D tiruyi patkern ayd artapatkerman amanak kovum  eynharti diagram nanakvum  |D| -ov: Qani or |D| -i a oakanuyun erku angam oqr  D -i a oakanuyunic, apa n = 2 n = 3 depqerum eynharti diagram talis  akner patkeracum tiruyi masin: z 7→ r(z) = (|z1 |, . . . , |zn |)

§ 1.

Kompleqs tara uyun

eynharti tiruyneri parzaguyn rinakner karo en a{

ayel gund polidisk: Hetagayum menq ktesnenq, or eynharti tiruyner serto{ ren kapva en astianayin arqeri het: 5. r j a n a  t i r u y  n e r, oronc kentron a ∈ Cn ketn . dranq ayn tiruynern en, oronq amen mi z 0 keti het mekte parunakum en na ( ) z = a + (z 0 − a)eiθ = a1 + (z10 − a1 )eiθ , . . . , an + (zn0 − an )eiθ

tesqi bolor keter: Aknhayt , or eynharti tiruy rjana{  : Haka ak it  (tes xndir 1.8-): rjana tiruyner kapva en hamaser bazmandamneric kazmva arqeri het: 6. H a r t o g s i t i r u y  n e r : Nanakenq ze = (z1 , · · · , zn−1 ) , aynpes or z = (ez , zn ) : G ⊂ Cn tiruy kovum  Hartogsi tiruy zn = an hamaa uyan haruyamb, ee z ∗ ∈ G paymanic het um , or {(ez ∗ , zn ) : |zn −an | = |zn∗ −an |} rjanagi  s patkanum  G -in: Hartogsi tiruy kovum  lriv, ee nran patkanum  amboj {(ez ∗ , zn ) : |zn − an | 6 |zn∗ − an |} rjan: Hartogsi tiruyner kazmum en aveli layn das, qan n -rja{ na tiruyner. nrancic rjana linelu hatkuyun pa{ hanjvum  miayn st mek (tvyal depqum` verjin) o oxakani: n = 2 depqum Hartogsi tiruynern anvanum en na kisa{ rjana : Hartogsi tiruy miareqoren orovum  ir patke{ rov (Hartogsi diagram) z 7→ (ez , |zn |) artapatkerman ama{ nak nanakvum  | G : Hartogsi diagram n = 2 depqum talis  akner patkeracum tiruyi masin, orovhet na gtnvum  e aa tara uyunum: Ayd hangamanq erbemn gnum  xndir{ ner lu elis: Parzuyan hamar hetagayum ditarkelu enq Har{ togsi tiruyner, oronc hamaa uyan haruyun an = 0 haruyunn :

Glux Լ. Holomorf funkcianer

Hartogsi lriv tiruyi hamar sahmanenq R(ez ) = sup R , or{ te Supremum- vercva  st bolor ayn R -eri, oronc hamar bavararvum  {(ez , zn ) : |zn | 6 R} ⊂ G payman: R(ez ) - kovum  Hartogsi a avi, nra mijocov Hartogsi tiruy nerka{ yacvum  { } e |zn | < R(e G = z ∈ Cn : ze ∈ G, z)

tesqov, orte Ge -ov nanakva  G -i proyekcian Czn−1 ena{ e tara uyan vra: Hartogsi tiruyneri nkatmamb hetaqrqruyun payma{ navorva  nranov, or nranq Hartogsi arqeri hamar zugami{ tuyan tiruyner en: 7. X o  o v a k a  t i r u y  n e r : T ⊂ Cn tiruy kovum  xoovaka , ee amen mi z 0 keti het mekte na parunakum  het yal tesqi kamayakan ket` z = (z10 + iy1 , . . . , zn0 + iyn ),

−∞ < yj < ∞,

j = 1, . . . , n :

Xoovaka tiruy kareli  nerkayacnel T = B × Rny tesqov, orte B -n Rnx irakan tara uyan in-or tiruy  (xoova{ ka tiruyi himq): Aknhayt , or xoovaka tiruy liovin orovum  ir himqov: Haaxaki gtagor vum  na het yal grela ` T = B + iRny = x + iy : x ∈ B ⊂ Rnx , y ∈ Rny :

Miaa depqum xoovaka tiruyner klinen {(x, y) : a < x < b, −∞ < y < ∞}

erter, inpes na {(x, y) : x > a} {(x, y) : x < a} kisahar{ uyunner: Nenq, or φ : zk 7→ ezk , k = 1, . . . , n artapatkerum  a o{ xum  T xoovaka tiruy D = φ ◦ T eynharti tiruyi,

§ 2.

Astianayin arqer

nd orum B himqin hamapatasxanum  |D| eynharti dia{ gram: 8. G o r e n u u c i k t i r u y  n e r : Ow ucik tiruyi sahmanumneric mek het yaln ` D ⊂ Rn tiruy kovum 

u ucik, ee amen mi ezrayin x0 keti hamar goyuyun uni ayd ketov ancno tiruyi het hatvo hiperharuyun (henman hiperharuyun): Ays sahmanman mej hiperharu{

yunneri oxaren kareli  vercnel aveli ca r a oakanu{ yun (rinak, n−2 ) uneco haruyunner, stacva tiruy{ neri das klini aveli layn, qan sovorakan u ucik tiruyneri das: Ays datouyunner kira eli en na Cn -um: D ⊂ Cn tiruy kovum  g oren u ucik, ee amen mi ezrayin z 0 keti hamar goyuyun uni ayd ketov ancno ti{ ruyi het hatvo kompleqs hiperharuyun: §

2. Astianayin arqer

1. Zugamituan tiruy: Nanakenq Z -ov amboj veri

bazmuyun, Z+ -ov` o bacasakan amboj veri bazmuyun, Zn -ov` k = (k1 , . . . , kn ) , ki ∈ Z multiindeqsneri bazmuyun Zn+ -ov` ayn k -ern Zn -ic, oronc hamar ki ∈ Z+ , 1 6 i 6 n : Amen mi z ∈ Cn keti k multiindeqsi hamar nanakenq` z k = z1k1 z2k2 · · · znkn , k! = k1 ! k2 ! · · · kn !, |k| = k1 + k2 + · · · + kn :

Bazmapatik astianayin (kam parzapes astianayin) arq kovum  het yal tesqi arq` ∞ ∑ |k|=0

( )k ak z − z 0 ,

(1.4)

Glux Լ. Holomorf funkcianer

orte z, z 0 ∈ Cn , k ∈ Zn+ : Parzuyan hamar, inpes kanon, menq qnnarkelu enq z 0 = 0 depq, aysinqn, (1.4) arqi oxaren ditarkvelu  ∞ ∑

ak z k

(1.5)

|k|=0

tesqi arq: In-or  ov hamarakalenq (1.5) bazmapatik arqi andam{ ner ditarkenq stacva sovorakan arq: Nra hamar sah{ manva en zugamituyan gumari gaa arner, oronq, sa{ kayn, kaxva en hamarakalman eanakic: Erb or   ov hama{ rakalva arq zugamitum  bacarak, apa zugamituyun kaxva  hamarakalman eanakic bnakan  nra gumar ndunel orpes bazmapatik arqi gumar: Aysuhet (1.5) arqi zugamituyan masin xoselis menq nkati enq unenalu bacar{ ak zugamituyun: Dicuq G -n ayn bolor keteri bazmuyunn , orte zugami{ tum  (1.5) arq: Nra nerqin keteri G◦ bazmuyun kovum  arqi zugamituyan tiruy: Mer motaka npatakn ` nka{ ragrel zugamituyan tiruy: I tarberuyun miaa depqi, mit , or G -n G◦ -i nra ezri in-or enabazmuyan miavorum :

 r i n a k 1.1. Het yal arqi` ∞ ∑ |k|=0

z1k1 +1 z2k2 =

z1 (1 − z1 )(1 − z2 )

zugamituyan bazmuyun miavor bidiskn , or lracva  {z1 = 0} kompleqs uov: Bazmapatik astianayin arqeri, inpes miapatikneri hamar, tei uni Abeli eorem:

§ 2.

Astianayin arqer

 e o r e m 1.1 (Abel). Ee G◦ -n (1.5) arqi zugamituyan

tiruyn 

ζ ∈ G0 ,

apa

{z : |zk | 6 |ζk |, k = 1, . . . , n}

ak polidisk s patkanum  G0 -in: A p a c u y c: Bxum  ∞ ∑ |k|=0

|ak | |z k | 6

∞ ∑

|ak | |ζ k | < ∞

|k|=0

anhavasaruyunic: Abeli eoremic het um , or astianayin arqi zugamitu{ yan tiruy lriv eynharti tiruy : Menq ktesnenq, or ayd tiruyner unen s mi parz erkraa{

akan hatkuyun` nranq in-or imastov u ucik en: Hiecnenq, or E bazmuyun kovum  u ucik, ee kamayakan x′ x′′ keteri het mekte nran patkanum  na ayd keter miacno {x : x = tx′ + (1 − t)x′′ , 0 6 t 6 1} hatva : Dicuq D -n eynharti tiruy : Nanakenq ln |D| -ov |D| bazmuyan patker |z| 7→ ln |z| = (ln |z1 |, . . . , ln |zn |) artapat{ kerman amanak:

S a h m a n u m 1.2. eynharti

logarimoren u ucik, ee u ucik 

D tiruy ln |D| tiruy:

kovum 

S a h m a n u m 1.3. eynharti D tiruyi lriv, logarimoren

u ucik aan anvanenq amena oqr lriv, logarimoren u u{ cik eynharti tiruy, or parunakum  D -n:

 e o r e m 1.2. Astianayin arqi zugamituyan ti{ ruy lriv, logarimoren u ucik eynharti tiruy :

Glux Լ. Holomorf funkcianer

A p a c u y c: Dicuq z ′ , z ′′ ∈ G , orte G -n (1.5) arqi zu{

gamituyan bazmuyunn : Ayd depqum ∞ ∑

∞ ∑

|ak ||z ′ |k < ∞,

|k|=0

Dicuq

ln |z| =

= |z ′ |t |z ′′ |1−t :

|ak ||z ′′ |k < ∞ :

|k|=0

+ (1 − t) ln |z ′′ |, 0 < t < 1 , aysinqn |z| = kamayakan A > 0 B > 0 veri hamar

t ln |z ′ |

Qani or

At B 1−t 6 (A + B)t (A + B)1−t = A + B

(0 6 t 6 1),

apa, havi a nelov (1.6)-, kstananq` ∞ ∑

(1.6)

|ak ||z k | =

|k|=0

∞ ( ∑

|k|=0 ∞ ∑

|ak ||z ′ |k

)t (

|ak ||z ′ |k +

|k|=0

|ak ||z ′′ |k

∞ ∑

)1−t

|ak ||z ′′ |k < ∞ :

|k|=0

Owremn, ln |z| = ket, aysinqn ln |z ′ | ln |z ′′ |

ayraketerov hatva i kamayakan ket, patkanum  ln |G| baz{ muyan: Da nanakum , or ln |G| bazmuyun u ucik  , uremn, ln |G0 | -n u ucik tiruy : t ln |z ′ | + (1 − t) ln |z ′′ |

H e t a n q 1.1. Ee (1.5) arq zugamitum 

eynharti D tiruyum, apa na zugamitum  na D -i lriv, logari{ moren u ucik aani vra:

 r i n a k 1.2. Ee

∞ ∑

|k|=0

ak z1k1 z2k2

{z : |z1 | < 1, |z2 | < ∞}

∪

arq zugamitum 

{z : |z1 | < ∞, |z2 | < 1}

bazmuyan vra, apa ayn zugamitum  na amboj C2 -um: Parzvum , it  na haka ak` amen mi lriv, logarimo{

ren u ucik eynharti tiruy in-or astianayin arqi zugamituyan tiruy : Ays asti apacuyc kberenq errord

glxum:

§ 2.

Astianayin arqer

2. Gor akicneri gnahatum: Sovorakan (miapatik) as{ tianayin arqeri depqum haytni en Koii anhavasaruyun{ ∑∞ ner gor akicneri hamar. ee j=1 aj z1j arq zugamet 

rjanum vov, apa

nra gumari modul sahmana ak  M

|z1 | < r1

|aj | 6

M r1j

(1.7)

:

Nman pndum tei uni na bazmapatik arqeri hamar:  e o r e m 1.3. Dicuq (1.5) arq zugamet  U (0, r) poli{

diskum

∞ ∑

ak zk 6 M,

∀ z ∈ U (0, r) :

(1.8)

|k|=0

Ayd depqum`

M , rk

(1.9) A p a c u y c: Xmbavorelov (1.5) arqi andamnern st a an{ in o oxakanneri (da kareli  anel norhiv bacarak zuga{ mituyan), kstananq hajordakan arq. |ak | 6

∞ ∑

ak z k =

|k|=0

∞ ∑ k1 =0

∀ k ∈ Zn+ :

∞ ∑

z1k1

z2k2

k2 =0

∞ ∑

ak znkn :

kn =0

Yuraqanyur stacva arqi nkatmamb hajordabar kira elov (1.7) Koii anhavasaruyunner, stanum enq` ∞ ∑

z2k2

k3 =0

ak znkn 6

kn =0

k2 =0 ∞ ∑

∞ ∑

z3k3 · · ·

∞ ∑ kn =0

ak znkn 6

M r1k1

,

M r1k1 r2k2

,

Glux Լ. Holomorf funkcianer |ak | 6

M r1k1

=

· · · rnkn

M : rk

Gor akicneri hamar (1.9) gnahatakanner kareli  grtel:

 e o r e m 1.4. Dicuq (1.5) arq zugamet  D sahmana{

ak eynharti lriv tiruyum bavararum  (1.8) payma{ nin: Ayd depqum tei unen het yal anhavasaruyunner` |ak | 6

M , dk (D)

orte dk (D) = sup z k :

∀ k ∈ Zn+ ,

z∈D

A p a c u y c: Tiruyi lrivuyan paymanic het um , or na

anverj qanakov U (0, r) polidiskeri miavorum : st naxord eoremi, amen mi aydpisi polidiski hamar tei unen Koii M |ak | 6 k anhavasaruyunner: Fiqsa k -i hamar ayd an{ r havasaruyunneric ntrenq lavaguyn` M M M = = : k k sup r dk (D) r∈|D| r

|ak | 6 inf

r∈|D|

Astianayin arqi hatkuyunnern usumnasirelis dk (D) me uyunner katarum en kar or der: Orpes varuyun a a{ jarkum enq apacucel nranc het yal parz hatkuyunner: 1. Ee

D1 ⊂ D2 ,

apa

dk (D1 ) 6 dk (D2 ) :

2. Ee ρD = Dρ -n D tiruyi ρ gor akcov nmanadruyunn  0 keti nkatmamb, apa dk (Dρ ) = ρ|k| dk (D) : 3.

dkj (D) = [dk (D)]j ,

iv :

orte

j -n

amboj

o bacasakan

Kasenq, or D0 tiruy kompaktoren nka  D -i mej kgrenq D0 b D , ee D0 -n sahmana ak  D0 ⊂ D :

§ 2.

Astianayin arqer

L e m m a 1.1. Dicuq (1.5) arq zugamet D ∞

lriv tiruyum

D0 b D :

Ayd depqum

arq zugamet :

∑

|k|=0

eynharti |ak |dk (D0 ) vayin

A p a c u y c: Nanakenq D1 -ov D0 -n parunako amena{

oqr lriv n -rjana tiruy: Qani or D0 -n lriv , apa D1 b b D : Owremn goyuyun uni ρ > 1 iv aynpisin, or (D1 )ρ = = D2 b D , orte (D1 )ρ - D1 tiruyi nmanadruyunn : (1.5) arqi gumar anndhat  D2 -i vra , uremn, sahmana ak  in-or M vov: Kira elov eorem 1.4-, kstananq` |ak | 6 ∞ ∑

|ak |dk (D0 ) 6

|k|=0

6M

∞ ∑

|ak |dk (D1 ) 6

|k|=0 ∞ ∑

∞ ∑ |k|=0

dk (D1 ) = dk (D2 )

M , dk (D2 )

|k|=0

∞ ∑ dk (D1 ) = M |k| ρ dk (D1 ) ρ|k| |k|=0

 e o r e m 1.5. Orpeszi (1.5) arq zugamiti sahmana{

ak eynharti lriv D tiruyum, anhraet  bavarar, or ∞ ∑

ak dk (D)z k

(1.10)

|k|=0

arq zugamiti U miavor polidiskum: A n h r a  e  t u  y u n: Vercnenq ρ iv, 0 < ρ < 1

hamapatasxan

Dρ b D

∞ ∑ |k|=0

tiruy: st lemma 1.1-i

|ak |dk (Dρ ) =

∞ ∑ |k|=0

|ak |ρ|k| dk (D) =

Glux Լ. Holomorf funkcianer =

∞ ∑



m=0



∑



(1.11)

|ak |dk (D) ρ|k|

|k|=m

vayin arq zugamet : Aknhayt , or (1.11)-i aj mas (1.10) arqi hetqn  miavor polidiski ankyunag i vra: st Abeli eoremi, (1.10) arq zugamet  amboj polidiskum: B a v a r a r u  y u n: Ee (1.10) arq zugamet  amboj U polidiskum, apa ayn zugamet  na {z ∈ Cn : |z1 | = · · · = |zn | = ρ, 0 < ρ < 1}

ankyunag i vra: Het abar, zugamet  (1.11) vayin arq: Aysteic het um  (1.5) arqi zugamituyun Dρ -um, orovhet erb z ∈ Dρ , apa ∞ ∑

|ak z k | 6

|k|=0

Qani or D -um:

ρ ∈ (0, 1)

∞ ∑ |k|=0

|ak | sup z k = z∈Dρ

∞ ∑

|ak |dk (Dρ ) < ∞ :

|k|=0

kamayakan , uremn (1.5) arq zugamet 

Aym nermu enq mek o oxakani depqum zugamituyan a{

avi bazmaa nmanak: S a h m a n u m 1.4. r1, . . . , rn ver, orte ri > 0, i = 1, . . . , n , kovum en zugamituyan hamalu a aviner (1.5) arqi hamar, ee ayd arq zugamitum  U (0, r) , r = (r1 , . . . , rn ) polidiskum i zugamitum o mi uri polidiskum, or pa{ runakum  U (0, r) -: I tarberuyun miaa depqi, orte zugamituyan a a{ vi orovum  miareq, bazmaa depqum hamalu a avi{ ∞ ∑ ner orovum en o miareq: rinak, (z1 z2 )m arqi zugami{ m=0 tuyan hamalu a aviner kapva en r1 r2 = 1 havasaru{ yamb:

§ 2.

Astianayin arqer

Hamalu a avineri hamar goyuyun uni Koi-Hadamari bana i nmanak:  e o r e m 1.6. (1.5) arqi zugamituyan hamalu a{

aviner bavararum en het yal a nuyan` √

lim

|k|

|k|→∞

A p a c u y c: Teadrelov arqi mej xmbavorelov nra andamner, kstananq` ∞ ∑

am z

m

=

|m|=0

∞ ∑

=

k=0

z = rζ, ζ ∈ C

vera{

am rm ζ |m| =

|m|=0 ∞ ∑

(1.12)

|ak |rk = 1 :

 

∑

 am r m  ζ k =

|m|=k

∞ ∑

(1.13)

ck ζ k :

k=0

Qani or (1.5) arqi hamar r1 , . . . , rn ver hamalu a aviner en, apa ∞ ∑

am z m =

|m|=0

∞ ∑

am (rζ)m

|m|=0

arq zugamet , erb |ζ| < 1 taramet , erb Koi-Hadamari bana i, (1.13)-ic kstananq` lim

k→∞

√ k

|ck | = lim

k→∞

√∑ k

ζ > 1:

am rm = 1,

st (1.14)

|m|=k

nd orum ayd havasaruyun anhraet  bavarar, orpeszi r1 , . . . , rn ver linen zugamituyan hamalu a aviner: M{ num  cuyc tal, or verin sahmanner (1.12)-um (1.14)-um irar havasar en: Da miangamic er um  het yal anhavasaruyun{ neric` |am′ | 6

∑

|m|=k

am r m 6

(n + k − 1)! ′ |am′ |rm , (n − 1)! k!

Glux Լ. Holomorf funkcianer

orte m′ indeqsin hamapatasxanum  |am |rm tesqi amename

(n + k − 1)! gumarelin, erb |m| = k , isk iv gumarelineri nd{ (n − 1)! k! hanur qanakn : Nkatenq, or (1.12)- kareli  grel Φ(r1 , . . . , rn ) = 0 harabe{ rakcuyan tesqov, orn irenic nerkayacnum  arqi zugamitu{ yan tiruyi eynharti diagrami ezri havasarum:

§

3. Holomorf funkciayi sahmanum

S a h m a n u m 1.5. Kompleqsareq f funkcian kovum 

holomorf kam analitik Ω ⊂ Cn tiruyum, ee

(a) f - anndhat  Ω -um, (b) f - holomorf  st amen mi o oxakani: Aveli grit (b) payman nanakum  het yal` ee 1 6 k 6 n , apa

z∈Ω

fk (ζ) = f (z1 , . . . , zk−1 , zk + ζ, zk+1 , . . . , zn )

mek o oxakani funkcian holomorf  st ζ -i C haruyan vra zro keti or  rjakayqum: Hetaqrqir  ayn hangamanq, or (b) paymanic het um  (a)-n:

 e o r e m 1.7 (Hartogs). Ee f

yuraqanyur o oxakani D ⊂ apa na holomorf  D -um:

funkcian holomorf  st tiruyi bolor keterum,

Cn

Ayd eoremi apacuyc menq enq beri ayn pata ov, or min hima ka hamematabar mateli apacuyc: Nkatenq, or Hartog{ si eoremi nmanak irakan o oxakanneri hamar it :

§ 3.

Holomorf funkciayi sahmanum

 r i n a k 1.3. Het yal funkcian  

f (x, y) =

x2

 0,

xy , + y2

erb (x, y) ̸= (0, 0), erb (x, y) = (0, 0)

anverj diferenceli  st x -i fiqsa y -i depqum haka ak, bayc nuynisk anndhat  (0, 0) ketum: Holomorfuyan gaa ar sahmanvum  na kamayakan baz{ muyan vra. f - kovum  holomorf K bazmuyan vra, ee na holomorf  K -i in-or bac rjakayqum: Berva payman i kareli oxarinel holomorfuyamb K -in patkano yuraqan{ yur ketum: Ayd nrbuyun lav er um  het yal rinakic:

 r i n a k 1.4. Dicuq

K ⊂ C2 bazmuyun kazmva  B1 = {z : |z − (0, 1)| 6 1/2} B2 = {z : |z + (0, 1)| 6 1/2} ak gnderic u dranq miacno L = {z : z1 = 0, z2 = x2 , |x2 | 6 1/2} hatva ic: K -i vra oroenq het yal funkcian`   erb z ∈ B1 ,  z1 , f (z) = 0, erb z ∈ L,   −z , erb z ∈ B :

Aknhaytoren, f - anndhat  K -i vra amen mi z0 ∈ K keti hamar kareli  nel Uz0 rjakayq, ur f - arunakvum  orpes holomorf funkcia: Iroq, B1 -in patkano keteri hamar, ne{ ra yal B1 -i L -i hatman (0, 1/2) ket, orpes Uz0 kvercnenq or  gund, or i hatvum B2 -i het f - karunakenq havasar z1 -i: B2 -in patkano keteri hamar kkatarenq nman ka ucum, arunakelov f - orpes −z1 : Ev verjapes, L -i nerqin keteri hamar kvercnenq gnder, oronq en parunakum hatva i ay{ raketer f - karunakenq orpes nuynabar zro: Sakayn, miakuyan eoremic, or menq kapacucenq qi heto, het um

Glux Լ. Holomorf funkcianer

, or f - hnaravor  arunakel orpes holomorf funkcia K bazmuyan or  Ω kapakcva rjakayq: Iroq, nva eoremic het um , or Ω -um goyuyun uni holomorf funkcia, or Ω -in patkano mi gndi vra havasar  z1 -i, isk myusi vra` −z1 -i: 1. Formal a ancyalner: Holomorfuyan sahmanumic hete{ vum , or f = u + iv funkciayi hamar st amen o oxakani bavararvum en Koi{ imani paymanner.  ∂v ∂u   = ,  ∂xk ∂yk ∂v ∂u    =− , ∂yk ∂xk

(1.15) k = 1, . . . , n :

Koi{ imani paymanner harmar  grel ( ) ∂f 1 ∂f ∂f = −i , ∂zk 2 ∂xk ∂yk ( ) ∂f 1 ∂f ∂f = +i : ∂ z̄k 2 ∂xk ∂yk

(1.16)

formal a ancyalneri mijocov, oronq gtakar en na uri har{ cerum: Ayd depqum (1.15) 2n irakan havasarumneri hamakarg grvum  orpes n kompleqs havasarumneri hamakarg` ∂f = 0, ∂ z̄k

k = 1, . . . , n :

(1.16)-um masnakco me uyunner anvanum en formal a anc{ yalner na ayn pata ov, or dranq kareli  stanal formal ki{ ra elov bard funkciayi a ancman kanon` ∂f ∂f ∂xk ∂f ∂yk = + = ∂zk ∂xk ∂zk ∂yk ∂zk

(

∂f ∂f −i ∂xk ∂yk

) ,

§ 3.

Holomorf funkciayi sahmanum

nuyn  ov ∂f ∂f ∂xk ∂f ∂yk = + = ∂ z̄k ∂xk ∂ z̄k ∂yk ∂ z̄k

(

∂f ∂f +i ∂xk ∂yk

) ,

k = 1, . . . , n : Qani or formal a ancyalnern artahaytvum en sovorakan a ancyalneri mijocov g oren, nranc hamar mnum en marit a ancman bolor kanonner: Haaxaki da gnum  he{ tuyamb stugel tvyal funkciayi holomorf linel, inpes, ri{ nak, stor berva eoremum`

 e o r e m 1.8. Dicuq f - holomorf  D ⊂ Cn

tiruyum g -n holomorf vektor-funkcia  G ⊂ -um, nd orum g(G) ⊂ D : Ayd depqum h = f ◦ g bard funkcian holomorf  G -um: Cm

A p a c u y c: Havelov h funkciayi formal a ancyalnern

st

ζ̄k -i ( k = 1, . . . , n ),

stanum enq

∑ ∂f ∂g ∑ ∂f ∂ḡ ∂h = + = 0, ∂zj ∂ ζ̄k ∂ z̄j ∂ ζ̄k ∂ ζ̄k j=1 j=1 n

qani or st f -i

g -i

n

holomorfuyan

∂f =0 ∂ z̄j

∂g =0: ∂ ζ̄k

Ayspisov, g -n holomorf  st yuraqanyur o oxakani , baci dranic, na aknhaytoren anndhat : Aym nkatenq, or

f

funkciayi

n ∑ ∂f ∂f df = dxk + dyk ∂xk ∂yk k=1

Glux Լ. Holomorf funkcianer

a ajin diferencial kareli  grel kompleqs tesqov, gtvelov (1.16)-ic trohelov ayn erku masi. df =

n n ∑ ∑ ∂f ∂f ¯ dzk + dz̄k = ∂f + ∂f, ∂zk ∂ z̄k k=1

k=1

¯ -` hakaholomorf orte ∂f - df diferenciali holomorf, isk ∂f masern en: Holomorfuyan payman diferenciali terminnerov grvum  hakir` ¯ =0: ∂f ∂¯ -havasarumn : § 26 -um menq k a{

Sa ayspes kova , hamase

noananq ays havasarman anhamase tarberaki het: Aym nkatenq, or (1.15)-um masnakco erku funkcianer bavararum en vov 2n hat havasarumneri: Erb n > 1 , ayd havasarumneri hamakarg linum  gerorova . dranov  ba{ catrvum ayn hangamanq, or at o oxakani holomorf funk{ cianern unen yurahatkuyunner, oronq bnoro en mek o o{ xakani depqin: Ω tiruyum holomorf funkcianeri das nanakvum  O(Ω) : Qani or holomorf funkcianeri gumarn u artadryal noric holomorf en, apa O(Ω) -n ak : Ee f ∈ O(Ω) funkcian anndhat  Ω -i akman vra, apa parzvum , or n > 1 depqum oro tiruyneri hamar, rinak, bazmaglanneri, f - in-or imas{ tov holomorf  nuynisk tiruyi ezri vra: Ayd pndman grit  akerpum trvum  lemma 1.2-um, orte nanakumneri par{ zuyan hamar ditarkum enq miavor U n polidiski depq: L e m m a 1.2. Dicuq f ∈ O(U n) ∩ C(U n) ze ∈ U n−1 ket fiqsa

: Ayd depqum f (ez , ζ) funkcian patkanum  ∩ O(U ) C(U ) -in: A p a c u y c: Dicuq gze(ζ) = f (ez , ζn ) : Ee ze ∈ U n−1 , apa

lemmayi pndum het um  holomorfuyan sahmanumic: gtec{ nenq ze- U n−1 polidiski ezrin: Qani or f - anndhat  U n -i

§ 4.

Plyuriharmonik funkcianer

vra, apa hamapatasxan gze funkcianer zugamitum en ha{ vasaraa , st Vayertrasi eoremi, sahmanayin funkcian ∩ patkanum  O(U ) C(U ) -in: §

4. Plyuriharmonik funkcianer

Formal a ancyalneri (1.16) sahmanumic het um , or ee f - holomorf , apa` ∂ f¯ = ∂zk

(

∂f ∂f −i ∂xk ∂yk

(

)

=

∂f ∂ z̄k

A ancelov st zk -i f funkciayi havi a nelov (1.17)-, kstananq`

)

= 0, u=

k = 1, . . . , n :

(f + f¯)

∂u 1 ∂f = : ∂zk 2 ∂zk

(1.17)

irakan mas (1.18)

Stor kapacucenq, or holomorf funkcian uni bolor kargi a ancyalner, oronq irenc herin holomorf en: Isk aym gtvenq ayd astic u s mek angam a ancelov (1.18)- st z̄j -i, ksta{ nanq` ∂2u = 0, k, j = 1, . . . , n : (1.19) ∂zk ∂ z̄j

peratorneri gnuyamb ays paymanner kareli  grel aveli hakir tesqov: Iroq, qani or ∂

∂¯

¯ = ∂ ∂u

n ∑ k,j=1

∂2u , ∂zk ∂ z̄j

apa (1.19)- hamareq  ¯ =0 ∂ ∂u

(1.20)

Glux Լ. Holomorf funkcianer

paymanin:

S a h m a n u m 1.6. Irakan funkcian kovum  plyuri{

harmonik D tiruyum, ee ayn patkanum  C 2 (D) dasin

bavararum  (1.20) paymanin:

Anjatelov (1.19)-i irakan ke maser, kstananq plyuri{ harmonikuyan paymannern irakan koordinatnerov` ∂2u ∂2u + = 0, ∂xk ∂xj ∂yk ∂yj ∂2u ∂2u − = 0, ∂xk ∂yj ∂xj ∂yk

(1.21) k, j = 1, . . . , n :

Erkrord xmbi havasarumner k = j depqum, iharke, aknhayt en: Plyuriharmonik funkcianer kapva en mi qani o oxa{ kani holomorf funkcianeri het it aynpes, inpes haruyan depqum harmonik funkcianer kapva en mek o oxakani ho{ lomorf funkcianeri het:

 e o r e m 1.9.

tiruyum holomorf f funkciayi irakan ke maser plyuriharmonik en: D ⊂ Cn

A p a c u y c: u = Re f irakan masi hamar eorem as{

toren arden apacucva : Mnum  nkatel, or f -i het mekte holomorf  na −if funkcian or Im f = Re (−if ) :

Tei uni ays eoremi hakadar, bayc miayn lokal tarbe{ rakov:

 e o r e m 1.10. Ee u -n plyuriharmonik  z0 ∈ Cn

keti rjakayqum, apa goyuyun uni ayd ketum holomorf f funk{ cia, ori hamar u -n irakan mas :

§ 4.

Plyuriharmonik funkcianer

A p a c u y c: Ditarkenq ) n ( ∑ ∂u ∂u ω= − dxk + dyk ∂yk ∂xk k=1

diferencial  : Havelov nra diferencial, stanum enq` ) n ( ∑ ∂2u ∂2u dω = − (dxj ∧ dxk + dyj ∧ dyk ) + ∂xk ∂yj ∂xj ∂yk k,j=1 ) n ( ∑ ∂2u ∂2u + dxj ∧ dyk : + ∂xj ∂xk ∂yj ∂yk k,j=1

Aysteic er um , or dω = 0 payman, aysinqn` ω -i akuyun, hamareq  (1.21)-in, aysinqn` u -i plyuriharmonikuyan: In{ pes haytni , ak   lokal grit : Owremn, ω -n z 0 keti in-or rjakayqum uni naxnakan v(z) funkcia, or kareli  grel ∫z v(z) =

ω

(1.22)

z0

tesqov: norhiv ω -i akuyan (1.22) integral kaxva  in{ tegrman anaparhic: (1.22)-ic stanum enq dv = ω , kam  ∂v ∂u    ∂x = − ∂y k

k

 ∂v ∂u   = ∂yk ∂xk

paymanner, oronq vkayum en, or f = u + iv funkcian pat{ kanelov C 2 dasin, bavararum  Koi- imani paymannerin st yuraqanyur o oxakani: Owremn, na z 0 ketum holomorf , nd orum, u = Re f :

Glux Լ. Holomorf funkcianer Teadrelov (1.19)-i kam (1.21)-i mej ∂2u

=0 ∂zk ∂ z̄k ∂2u ∂2u + 2 =0 ∂x2k ∂yk

j = k,

kstananq`

kompleqs koordinatneri tesqov, kam irakan koordinatneri tesqov:

(1.23)

Ays havasarumnerin bavararo funkcian kovum  n -harmo{ nik ( n = 2 depqum` erkharmonik) funkcia: Stacva paymanne{ r nanakum en, or u -n harmonik  st amen mi zk o oxakani: Ee gumarenq (1.23) havasarumnern st k -i, apa ax ma{ sum kunenanq Puasoni ∆u perator kstacvi` ∆u = 0,

aysinqn, u -i harmonikuyan payman: Dicuq Pr (θ) =

1 − r2 1 − 2r cos θ + r2

Puasoni sovorakan korizn  miavor rjani hamar: Sahmanenq Puasoni koriz U n miavor polidiski hamar het yal  ov` P (z, ζ) = Pr1 (θ1 − φ1 ) · · · Prn (θn − φn ),

orte z ∈ U n , ζ ∈ T n , zj = rj eiθj , ζj = eiφj : Baci dranic, mn -ov nanakenq T n miavor tori vra Lebegi normavorva a : Ay{ sinqn` dmn = dφ1 · · · dφn : (2πi)n Puasoni koriz verlu vum  P (z, ζ) =

∞ ∑

|k |

r1 1 · · · rn|kn | eik·(θ−φ)

(1.24)

|k|=−∞

arqi, orte k ·θ = k1 θ1 +· · · +kn θn : Ee f - integreli  st mn a i, apa nra hamar sahmanvum  P [f ] Puasoni integral. ∫

P [f ](z) =

f (ζ)P (z, ζ) dmn (ζ), Tn

z ∈ Un :

§ 4.

Plyuriharmonik funkcianer

Inpes haytni , polidisk patkanum  ayn tiruyneri da{ sin, oronc hamar Dirixlei xndir lu eli 1 : Da nanakum , or amen mi f funkcia, or anndhat  U n -i ezri vra, hnaravor  anndhatoren arunakel polidiski akman vra aynpes, or na lini U n -um harmonik: Ee f - arunakenq o e ezric, ayl henqic, apa anndhat arunakuyun kareli  katarel n -har{ monik funkcianeri dasum: Aysinqn, marit  het yal eorem:

 e o r e m 1.11.

Dicuq f - kamayakan funkcia , orn anndhat  -i vra: Ayd depqum u(z) = P [f ](z) funkcian U n -um n -harmonik , anndhat  U n -i akman vra henqi vra hamnknum  f -i het: Tn

A p a c u y c: Teadrelov (1.24) verlu uyun Puasoni in{

tegrali mej andam a andam integrelov, kstananq` P [f ](z) =

∞ ∑

|k | fb(k)r1 1 · · · rn|kn | eik·θ ,

(1.25)

|k|=−∞

orte

fb(k)

ver f -i Furyei gor akicnern en` fb(k) =

∫

ζ̄ k dmn (ζ),

k ∈ Zn :

Tn

(1.25) nerkayacumic anmijapes het um , or u(z) - n -harmo{ nik : Hiecnenq, or henqi vra orova T (ζ) funkcian kovum  e ankyunaa akan bazmandam, ee na verjavor vov eik·θ qs{ ponentneri g ayin kombinacia : E ankyunaa akan bazman{ dami depqum (1.25) verlu uyan mej masnakcum en verjavor vov andamner, usti eoremi pndum nra hamar aknhayt :

Te|s, rinak, Axler Տե., Bօսrմօո P., Խոmey Ն. Ha.բոոic fuոcեiոո եhօո.կ, Տքrոոger-Verlոg, NeՇ Yօrk, Iոօ., 2001, j 228{230:

Glux Լ. Holomorf funkcianer

depqum Feyeri haytni eorem pndum , or amen mi annd{ hat funkcia havasaraa motarkvum  e ankyunaa akan bazmandamnerov: ndhanur depqum ayd ast s marit  st Stoun-Vayertrasi eoremi: Dicuq Tm (ζ) e ankyunaa{

akan bazmandamneri hajordakanuyun havasaraa T n -i vra zugamitum  f -in: Inpes het um  maqsimumi skzbunqic, Puasoni integralneri P [Tm ](z) hamapatasxan hajordaka{ nuyun havasaraa zugamitum  amboj U n -i vra, qani or ayd sahman havasar  P [f ] -n, apa u(z) = P [f ](z) funkcian anndhat  U n -um: n=1

§

5. Holomorf funkciayi zroner

Mek o oxakani funkcianeri tesuyunum haytni  het { yal pndum`

 e o r e m 1.12. Ee f (z) funkcian holomorf  a ketum, f (a) = 0

f ̸≡ 0 ,

apa a -i in-or rjakayqum f (z) = (z − a)p · h(z),

orte p > 1 amboj iv , isk h(z) - holomorf  uni ayd rjakayqum:

zroner

Ays eorem mi o oxakani funkcianeri zroneri masin ta{ lis  het yal informacian` 1. zroner mekusacva en, 2. f -i zroner hamnknum en (z − a)p funkciayi zroneri het, nd orum, p -n kovum  zroyi karg:

§ 5.

Holomorf funkciayi zroner

Mi qani o oxakani funkcianeri zroner hetazotelis kar or der uni het yal eorem, orn apes ndhanracnum  naxord eoremi pndum:

 e o r e m 1.13 (Vayertrasi naxapatrastakan eorem).

Dicuq f funkcian holomorf  a ∈ Cn keti rjakayqum, f (a) = 0 , bayc f (e a, zn ) ̸≡ 0 : Ayd depqum a -i in-or V rja{ kayqum f -n uni het yal nerkayacum` f (z) = W (z) · h(z),

orte h(z) - holomorf  ayd rjakayqum zro i da num, isk W (z) - ayspes kova , Vayertrasi ps dobazmandam ` W (z) = (zn − an ) + p

p−1 ∑

cj (e z )(zn − an )j :

j=0

Ayste cj (ez ) gor akicner holomorf en Ve -um, nd orum Ve nanakum  V -i proyekcian Cn−1 enatara uyan vra, ze cj (e a) = 0 , j = 0, . . . , p − 1 : A p a c u y c: ndhanruyun xaxtelov, karo enq ena{

drel a = 0 : st miakuyan eoremi mek o oxakani funkcia{ neri hamar, goyuyun uni rn > 0 aynpisin, or f (e0, zn ) ̸= 0 , erb 0 < |zn | 6 rn , isk f -i anndhatuyan norhiv kgtnvi aynpisi mi polidisk Ve (e0, r) , or f (e z , zn ) ̸= 0,

erb

ze ∈ Ve

|zn | = rn :

Amen mi fiqsa ze0 ∈ Ve keti hamar f (ez 0 , zn ) funkciayi zroneri qanak Vn = {zn : |zn | < rn } rjanum havasar  k1 =

2πi

∫ ∂Vn

∂ f (e z 0 , zn ) ∂zn dzn : f (e z 0 , zn )

(1.26)

Glux Լ. Holomorf funkcianer

Iroq, (1.26)-i ax mas ndunum  miayn amboj areqner anndhat  st ze0 -i Ve -um, usti na nuynabar hastatun : Erb ze0 = e0 , na havasar  f (e0, zn ) funkciayi zroyi kargin zn = 0 ketum, aysinqn` k -in: Aym fiqsenq ze ∈ Ve nanakenq zn(k) = zn(k) (ez ) , k = 1, . . . , p , f (e z , zn ) funkciayi zroner Vn rjanum ka ucenq zn -i nkat{ mamb P (z) =

p [ ∏

] zn − zn(k) (e z ) = znp + cp−1 (e z )znp−1 + · · · + c0 (e z)

(1.27)

k=1

bazmandam, ori armatner nva zronern en: Cuyc tanq, or nra gor akicner holomorf en Ve -um: Iroq, kamayakan ω(zn ) funkciayi hamar, or holomorf  V n -um, st argumenti nd{ hanracva skzbunqi` p ∑ k=1

ω(zn(k) ) = 2πi

∫

∂ f (e z , zn ) ∂z ω(zn ) n dzn , f (e z , zn )

∂Vn

orteic er um , or ax masum gtnvo gumarnern st ze-i holomorf funkcianer en Ve -um: Ayste havi  a nva ayn han{ gamanq, or f - tarber  zroyic, erb ze ∈ Ve zn ∈ ∂Vn : Tea{ drelov ω(zn ) = znj , j = 1, . . . , p , stanum enq, or (1.27) bazmanda{ mi armatneri j -rd astiani gumarner holomorf funkcianer en st ze-i Ve -um: Inpes haytni  hanrahavic, bazmandami gor akicner ayd gumarneric acional funkcianer en, orteic bxum  nranc holomorfuyun: Erb ze = e0 , bazmandami bolor gor akicner zro en da num, usti bolor ck (e0) = 0 : Aynuhet ` h(z) =

f (z) P (z)

funkcian fiqsa ze ∈ Ve depqum holomorf  st zn -i Vn rja{ num aynte zroner uni, qani or P -n f - unen mi nuyn

§ 5.

Holomorf funkciayi zroner

kargi mi nuyn zn(k) (ez ) zroner: Owremn, kamayakan ze ∈ qum h - nerkayacvum  Koii integralov st zn -i` h(z) =

2πi

∫

∂Vn

Ve

dep{

f (e z , ζn ) dζn : P (e z , ζn ) ζn − zn

Dranov na sahmanvum  V = Ve × Vn bazmuyan na ayn kete{ rum, orte P = 0 : Qani or P ̸= 0 ∂Vn -i vra, apa aj mas, usti h -, holomorf en st ze-i: st Hartogsi eoremi h(z) funkcian holomorf  V polidiskum: Vayertrasi naxapatrastakan eoremi areq kayanum  nranum, or na uyl  talis hetazotel holomorf funkciayi zroner orpes ps dobazmandami zroner funkciayi zroyakan keti rjakayqum, aysinqn` lokal: Ayl ba erov asa , holomorf funkciayi zroneri hetazotman xndir bervum  ps dobaz{ mandami zroneri hetazotman, aysinqn, st mi o oxakani bazmandami, ori gor akicner holomorf en st mnaca

o o{ xakanneri: Aym cuyc tanq, e inpes kareli  gtagor el Vayertrasi eoremn anbacahayt funkcianeri masin eoremi kompleqs tar{ ∂f berak stanalu hamar: Dicuq f (a) = 0 ̸= 0 : Kira{ ∂zn z=a

elov ayd eorem (nd orum p = 1 ), stanum enq, or f (z) = 0 havasarum hamareq  P (z) = zn −an +c0 (ez ) = 0 havasarman, or lu eli  st zn -i, zn = an − c0 (ez ) funkcian holomorf  st mnaca

o oxakanneri: S a h m a n u m 1.7. Dicuq D ⊂ Cn A ⊂ D : A -n kovum  analitik bazmuyun D -um, ee kamayakan a ∈ D keti hamar goyuyun unen U ∋ a rjakayq U -um aynpisi holomorf f1 , . . . , fN funkcianer, or A

∩

U = {z ∈ U : f1 (z) = · · · = fN (z) = 0} :

(1.28)

Glux Լ. Holomorf funkcianer

Ayspisov, analitik bazmuyun lokal sahmanvum  orpes verjavor vov holomorf funkcianeri ndhanur zroneri baz{ muyun:  e o r e m 1.14. D -um A analitik bazmuyun ( A ̸= D )

ak , amenureq nosr  nra D \ A lracum kapakcva : A p a c u y c: Dicuq am ∈ A , lim am = a ∩ a ∈ D . petq 

apacucel, or a ∈ A : a keti U rjakayqum A U bazmuyun trvum  (1.28) paymanov fm funkcianeri anndhatuyunic het um , or a ∈ A : Enadrenq A -i nerqin keteri A◦ enabazmuyun datark : Cuyc tanq, or A◦ -n ak  D -um: Dicuq am ∈ A◦ , lim am = a a ∈ D . petq  apacucel, or a ∈ A◦ : Da het um  ayn banic, or U rjakayqum A -n oroo fk funkcianer havasar en zroyi U -i bac enabazmuyan vra , st miakuyan eoremi, fk ≡ 0 , k = 1, . . . , N amboj U -um, aysinqn, U ⊂ A◦ : Ayspisov, A◦ -n D tiruyi o datark, miaamanak bac ak enabazmuyun , uremn, A◦ = D : Verjin pndum apacucelu hamar bavakan  apacucel, or amen mi a ∈ A ket uni aynpisi U kapakcva rjakayq, or U \A bazmuyun kapakcva : Dicuq a ∈ A kamayakan ket , U -n nra or  u ucik rjakayq  z 0 -n u z 1 - kamayakan keter en (U \ A) -ic: Nanakenq { } G = ζ ∈ C : ζz 0 + (1 − ζ)z 1 ∈ U -ov

U -i

hatum z 0 z 1 keter miacno kompleqs ui het: G -n har u ucik tiruy : U rjakayqum A -n oroo fk funkci{ aneri mej kgtnvi aynpisin, or gk (ζ) = fk (ζz 0 + (1 − ζ)z 1 ) ̸≡ 0,

het abar,

{ } H = ζ ∈ C : ζz 0 + (1 − ζ)z 1 ∈ A

§ 6.

Koii bana  nra parzaguyn kira uyunner

bazmuyun diskret : Ayd pata ov G \ H bazmuyun ka{ pakcva , qani or na parunakum  ζ0 = 0 u ζ1 = 1 keter, apa goyuyun uni ayd keter miacno γ : I 7→ G \ H anndhat ( ) kor: Ayd depqum t 7→ γ(t)z + 1 − γ(t) z 1 funkcian (U \ A) bazmuyan vra oroum  kor, or miacnum  z 0 -n z 1 -:

§

6. Koii bana  nra parzaguyn kira uyunner Dicuq D = {z ∈ Cn : z1 ∈ D1 , . . . , zn ∈ Dn }

bazmaglan  Γ = {z ∈ Cn : z1 ∈ ∂D1 , . . . , zn ∈ ∂Dn }

nra henqn , orte D1 , . . . , Dn har tiruyner sahmana ak en unen ktor a ktor oork ezr: ∩

 e o r e m 1.15.

Kamayakan f ∈ O(D) C(D) funkcia cankaca z ∈ D ketum nerkayacvum  f (z) = (2πi)n

∫ Γ

f (ζ) dζ1 · · · dζn (ζ1 − z1 ) · · · (ζn − zn )

(1.29)

Koii bazmapatik integralov: A p a c u y c: Kira elov Koii integralayin bana  f (ez , zn )

funkciayi nkatmamb st verjin o oxakani, kstananq f (e z , zn ) = 2πi

∫ ∂Dn

f (e z , ζn ) dζn : ζn − zn

Glux Լ. Holomorf funkcianer Nuyn  ov

f (e z , ζn ) -i

f (e z , ζn ) =

hamar

2πi

∫ ∂Dn

f (z1 , . . . , zn−2 , ζn−1 , ζn ) dζn−1 : ζn−1 − zn−1

Teadrelov ays havasaruyun naxordi mej arunakelov da{ touyunner, kstananq` f (z) = (2πi)n

∫

∂D1

dζ1 ζ1 − z1

∫

∂D2

dζ2 ζ2 − z2

∫

∂Dn

f (ζ1 , . . . , ζn ) dζn : ζn − zn

Stacva hajordakan integral, st Fubinii eoremi, hava{ sar  n -patik integralin, or masnakcum  (1.29) integralayin nerkayacman aj masum: Hetagayum (1.29) bana  grelu enq f (z) = (2πi)n

∫ Γ

f (ζ) dζ (ζ − z)I

(1.30)

hakir tesqov, orte dζ = dζ1 · · · dζn , I = (1, . . . , 1) , uremn, st mer nduna nanakumneri, (ζ − z)I = (ζ1 − z1 ) · · · (ζn − zn ) : D i t o  u  y u n 1.1. (1.30) bana  apes tarbervum  hamapatasxan bana ic mek kompleqs o oxakani funk{ cianeri hamar: Tvyal depqum (1.30) bana  verakangnum  funkciayi areqner tiruyi nersum st ir areqneri o e amboj ∂D ezri vra, ayl miayn nra n a ani masi` henqi vra: Myus komic, n = 1 depqum Koii bana  it  ktor a

ktor oork ezrov kamayakan tiruyi hamar, aynin mi qani

o oxakani funkcianeri depqum tei uni miayn bazmaglan{ neri hamar, aysinqn, bavakanin ne dasi hamar, it , rinak, gndi depqum: orrord glxum menq kstananq integra{ layin nerkayacumner tiruyneri aveli layn dasi hamar:

§ 6.

Koii bana  nra parzaguyn kira uyunner

Parametric kaxva integralneri masin ndhanur eorem{ neric het um , or (1.30)- kareli  a ancel integrali nani tak menq stanum enq k! ∂ |k| f (z) = k ∂z (2πi)n

∫ Γ

f (ζ) dζ . (ζ − z)k+I

(1.31)

Koii bana , inpes miaa depqum, hnaravoruyun  ta{ lis stanalu verlu uyun astianayin arqi: Hakiruyan hamar nanakenq { } G = z ∈ Cn : |zj − zj0 | < rj , j = 1, . . . , n

kentron polidisk,

r = (r1 , . . . , rn ) vektorakan bazmaa avi uneco Γ -ov` nra henq: ∩ 1.16 Ee f ∈ O(G) C(G) , apa cankaca z ∈ G

z0

eorem

.

ketum na nerkayacvum  f (z) =

∞ ∑

( )k ak z − z 0

(1.32)

|k|=0

astianayin arqov, ori gor akicner orovum en ak = (2πi)n

∫ Γ

f (ζ) dζ (ζ − z 0 )k+I

bana ov: A p a c u y c: Koii koriz verlu enq bazmapatik erkra{

a akan progresiayi gumari` I

(ζ − z)

=

(ζ −

z 0 )I

) ( )= ·( z1 − z1 zn − zn0 1− ··· 1 − ζn − zn0 ζ1 − z10

Glux Լ. Holomorf funkcianer

)k ∞ ( ∑ z − z0 = : (ζ − z 0 )I |k|=0 ζ − z 0

Ayspisov` (ζ − z)I

=

∞ ∑

(

z − z0

)k

(ζ − z 0 )k+I |k|=0

:

Nkatenq, or ays verlu uyun zugamitum  havasaraa st f (ζ) ζ -i, Γ -i vra: Bazmapatkelov ayn -ov andam a andam (2πi)n integrelov, stanum enq eoremi pndum: D i t o  u  y u n 1.2. eoremi pndum it linelu hamar

bavakan  pahanjel, or f ∈ O(G) : Iroq, kamayakan z ket patkanum  in-or G′ ⊂ G polidiski mnum  kira el eo{ rem 1.16- G′ -um:

 e o r e m 1.17. Ee f ∈ O(G) , apa cankaca z ∈ G ketum ayd funkcian uni bolor kargi masnaki a ancyalner, oronq s patkanum en O(G) -in: A p a c u y c: st eorem 1.16-i cankaca z ∈ U ketum

f -

nerkayacvum  (1.32) astianayin arqi tesqov: Qani or ayd arqi andamner kareli  xmbavorel st a anin o oxa{ kanneri astianneri, apa, gtvelov mi o oxakani astia{ nayin arqi hatkuyunneric, stanum enq, or f -i bolor masnaki a ancyalner nerkayacvum en astianayin arqi tesqov: In{ pes het um  (1.12)-ic, andam a andam a ancva arqer U -i vra zugamet en, uremn` dranc gumarnern anndhat en norhiv havasaraa zugamituyan U -i kompakt enabazmuyun{ neri vra: Kareli  apacucel uri eanakov, gtvelov a ancyalneri hamar stacva (1.31) bana ic parametric kaxva inte{ gralneri ndhanur hatkuyunneric:

§ 6.

Koii bana  nra parzaguyn kira uyunner

1. Vayertrasi eorem: Nax holomorf funkciayi a an{ cyalneri hamar stananq gnahatakanner ver ic:  e o r e m 1.18. Dicuq f funkcian holomorf sahma{

na ak  Ω tiruyum: Cankaca M b D kompakt ena{ bazmuyan k = (k1 , . . . , kn ) multiindeqsi hamar goyuyun uni C = C(M, k) hastatun, aynpisin, or max z∈M

∂ |k| f (z) 6 C max |f (z)| : z∈Ω ∂z k

(1.33)

A p a c u y c: Bavakan  apacucel kamayakan z 0 ∈ Ω keti

rjakayqum: Vercnenq U = U (z 0 , r) b Ω polidisk Koii integralayin bana ic a ancyalneri hamar` k! ∂ |k| f (z) = ∂z k (2πi)n

∫ Γ

gtvenq

f (ζ) dζ , (ζ − z)k+I

orte Γ -n U -i henqn : Aysteic stanum enq` ∂ |k| f (z) k! ∂z k (2π)n

(2π)n r1 · · · rn · maxζ∈Ω |f (ζ)| , min |ζ1 − z1 |k1 +1 · · · |ζn − zn |kn +1

z∈M, ζ∈Γ

orteic het um  (1.33)-: Kira elov (1.33)- fj − f tarberuyan nkatmamb, stanum enq het yal pndum:  e o r e m 1.19 (Vayertras). Dicuq fj ∈ O(D) hajorda{ kanuyun zugamitum  f funkciayin havasaraa D -i kompakt enabazmuyunneri vra: Ayd depqum f ∈ O(D) , , baci dranic, bolor k = (k1 , . . . , kn ) -i hamar ∂ |k| fj (z) ∂ |k| f (z) = j→∞ ∂z k ∂z k lim

havasaraa amen mi M b D vra:

Glux Լ. Holomorf funkcianer

2. Miakuyan eorem: Mi qani o oxakani depqum miaa tesuyan miakuyan eorem, orum ditarkvum  funk{ ciayi zroneri sahmanayin ket, tei uni: Dranum kareli  hamozvel f (z1 , z2 ) = z1 · z2 ̸≡ 0 funkciayi rinaki vra, or zro  da num {z : z1 = 0} {z : z2 = 0} erku kompleqs uineri vra: marit  ayd eoremi aveli uyl tarberak:

 e o r e m 1.20. Ee f - holomorf  Ω ∈ Cn

tiruyum f = 0 Ω -i o datark bac enabazmuyan vra, apa f ≡ 0 Ω -um: A p a c u y c: Nanakenq ω -ov {z ∈ Ω : f (z) = 0} bazmu{

yan ners: Ayd depqum ω -n Ω -i bac enabazmuyun : Dicuq zk ∈ ω zk → z ′ ∈ Ω : Qani or zk keterum f - havasar  zroyi ir bolor a ancyalneri het mekte, apa norhiv anndha{ tuyan, nranq havasar en zroyi na z ′ ketum: Stacvec, or ω -n ak  Ω -um: Inpes haytni , kapakcva bazmuyan bac miaamanak ak o datark enabazmuyun hamnknum  nra het, usti ω = Ω : 3. Maqsimumi skzbunq:

 e o r e m 1.21. Ee f ∈ O(D)

in-or a ∈ D ketum, apa

|f | - uni lokal maqsimum f (z) ≡ const amboj D -um:

A p a c u y c: a ketov ancno cankaca

{z ∈ Cn : z = a + ωζ},

ω ∈ Cn ,

ζ∈C

kompleqs ui vra f funkciayi hetq, aysinqn` gω (ζ) = f (a + + ωζ) -n, mek o oxakani holomorf funkcia , ori modul ζ = = 0 nerqin ketum uni maqsimum, uremn, gω (ζ) ≡ cω : Qani or gω (0) = f (a) kaxva  ω -ic , myus komic, cankaca erku ketov ancnum  kompleqs ui (varuyun 1.1), apa stanum

§ 6.

Koii bana  nra parzaguyn kira uyunner

enq, or a -i rjakayqum f (z) ≡ f (a) : f (z) ≡ const amboj D -um:

st miakuyan eoremi,

Ee D -n sahmana ak  f - holomorf  D -um, apa |f | - ir maqsimumn ndunum  ∂D ezri vra: Sakayn n > 1 depqum Cn tara uyunum hnaravor en D tiruyner, oronc hamar ayd maqsimum ndunum  o e amboj ezri, ayl nra mi masi vra:

S a h m a n u m 1.8. Sahmana ak D tiruyi hamar B(D) ⊂

⊂ ∂D

ak bazmuyun kovum  Bergmani ezr, ee.

1) amen mi

f ∈ O(D)

funkciayi hamar

max |f (z)| = max |f (z)|, z∈B(D)

z∈D

2) cankaca ayl bazmuyun, or bavararum  1) paymanin, parunakum  B(D) -n: S(D) bazmuyun kovum  ilovi ezr, ee nva paymanner ∩ bavararvum en bolor f ∈ O(D) C(D) funkcianeri hamar: {

∩

}

Qani or O(D) ⊂ O(D) C(D) , apa parz , or B(D) ⊂ S(D) : Erb D -n gund , apa Bergmani ilovi ezrer hamnknum en nra topologiakan ezri (aysinqn` sferayi) het: Erb D -n polidisk , apa Bergmani ilovi ezrer hamnknum en arden o e nra topologiakan ezri, ayl henqi het (tes xndirner 2.12 2.13): Hnaravor en depqer, erb Bergmani ilovi ezrern iraric tarbervum en (xndir 2.14): 4. Liuvili eorem: Inpes

het yal pndum:

 e o r e m 1.22 (Liuvil).

miaa depqum, tei uni

Ee Cn -um amboj funkcian sahmana ak , apa na nuynabar hastatun :

Glux Լ. Holomorf funkcianer A p a c u y c: Enadrenq |f (z)| 6 M

tianayin arqi`

∞ ∑

f (z) =

verlu enq f - as{

ak z k :

|k|=0

Ays arq zugamet  amboj saruyunner`

Cn -um,

|ak | 6

usti Koii (1.9) anhava{

M r|k|

marit en bolor r -eri hamar: Ayste r = r1 = · · · = rn : Ancnelov sahmani, erb r → ∞ , stanum enq, or ak = 0 , ee |k| = k1 + · · · + + kn > 0 , aysinqn, f (z) ≡ a0 : §

7. Ayl arqer

1. Lorani arqer: Aym ditarkenq L o r a n i  a r q i

bazmaa nmanak:

 e o r e m 1.23. Amen mi f

funkcia, or holomorf 

Π(r, R) = {z ∈ Cn : rk < |zk | < Rk , k = 1, . . . , n}

rjanayin akneri dekartyan artadryali vra, nerka{ yacvum  f (z) =

∞ ∑

ck z k

(1.34)

|k|=−∞

bazmapatik Lorani arqi tesqov, ori gor akicnern en` ck = (2πi)n

orte γ = γ1 × · · · × γn , rj < ρj < Rj , j = 1, . . . , n :

∫

f (ζ) dζ , k+I γ ζ { } γj = ζ : ζj = ρj eit , 0 6 t 6 2π ,

§ 7.

Ayl arqer

Apacuyc katarvum  nuyn meodov, in miaa i depqum: Tei uni aveli ndhanur pndum:

 e o r e m 1.24.

Ee f - holomorf  eynharti D ti{ ruyum, apa na D -i nersum nerkayacvum  bacarak havasaraa zugamet Lorani (1.34) arqov: A p a c u y c: D tiruy kareli  nerkayacnel orpes an{

verj qanakov Π(r, R) tesqi bazmaglanayin tiruyneri miavo{ rum: Drancic yuraqanyurum st eorem 1.23-i f (z) - nerka{ yacvum  Lorani (1.34) arqov: Verlu uyan miakuyunic hete{ vum , or Π1 (r′ , R′ ) Π2 (r′′ , R′′ ) hatumneri vra hamapatasxan Lorani arqer hamnknum en: Qani or D tiruyi cankaca

erku ket kareli  miacnel irar het hatvo Π(r, R) tesqi baz{ maglannerov, apa, havi a nelov Lorani arqi miakuyan hat{ kuyun, amboj D -um f - nerkayacvum  mi nuyn Lorani ar{ qov: Ays eoremic orpes het anq stacvum  het yal pndum:

 e o r e m 1.25 (gndayin erti masin). Ee f - holomorf  { } D = z ∈ C2 : r < |z| < R

gndayin ertum, apa ayn analitikoren arunakvum  { } B(0, R) = z ∈ C2 : |z| < R

gndi mej: A p a c u y c: st eorem 1.24-i f - D -um nerkayacvum 

(1.34) arqov: nakum en

D -um

kgtnven erku bazmaglanner, oronq paru{

{ } z ∈ C2 : z1 = 0

{ } z ∈ C2 : z2 = 0

Glux Լ. Holomorf funkcianer

koordinatayin haruyunneri keter: Ev uremn, ayd bazma{ glannerum f funkcian nerkayacno Lorani arq i karo une{ nal z2 -i hamapatasxanabar z1 -i bacasakan astianne{ rov gumareliner: Lorani arqi verlu uyan miakuyunic he{ t um , or ayd arq eylori arq : st Abeli eoremi na zugamitum  na B(0, R) gndum nra gumar talis  pa{ hanjveliq arunakuyun: Parz , or Lorani (1.34) arqi zugamituyan tiruy eyn{ harti tiruy , nd orum, na bavararum  s mi lracuci paymani: Ee na parunakum  or  z 0 ket zk0 = 0 koordinatov, apa (1.34) verlu uyan mej bacakayum en ayd koordinati ba{ casakan astianner, aysinqn, zk -i nkatmamb (1.34)- eylori arq : Owsti Lorani arqi zugamituyan tiruy fiqsa k -i hamar kam i hatvum zk = 0 haruyan het, kam l amen mi z 0 keti het mekte parunakum  bolor ayn z keter, oronc hamar |zk | 6 |zk0 | , isk mnaca koordinatner nuynn en, in z 0 -in: Ayd{ pisi tiruy kovum  eynharti kisalriv tiruy: 2. Hartogsi arqer: Andrada nanq ayl tipi arqeri, oron{

cic amenakar or H a r t o g s i  a r q n  : Aydpes  kovum ∞ ∑

gm (e z )(zn − an )m

m=0

tesqi arq, ori gm gor akicner holomorf funkcianer en:  e o r e m 1.26. Ee f funkcian holomorf  Hartogsi lriv

tiruyum, ori hamar {zn = 0} -n hamaa uyan har{ uyun , apa na nerkayacvum  D -i nersum bacarak havasaraa zugamet D

f (z) =

∞ ∑ m=0

gm (e z )znm

(1.35)

§ 7.

Ayl arqer

Hartogsi arqov, orte gm funkcianer holomorf en ayd tiruyi Cn−1 -i vra De proyekciayum: A p a c u y c: Dicuq a ∈ D : Vercnenq bavakanaa

oqr

polidisk

e = {e U z : |zi − ai | < ri , i = 1, . . . , n − 1} ⊂ Cn−1 e × Un po{ Un = {zn : |zn | < R} rjan aynpisiq, or U = U lidiskn nka lini D -i mej: Da hnaravor  anel norhiv D -i lrivuyan: Aynuhet f - D -um verlu enq astianayin arqi ∞ ∑

′

ak (e z− e a) k znkn :

|k|=0

Xmbavorenq ays arqi andamnern st zn -i astianneri, dranic nra gumar i oxvi bacarak zugamituyan norhiv: Kstac{ vi (1.35) tesqi arq, ori gm gor akicner holomorf en Ue -um orpes astianayin arqi gumarner: U tesqi polidiskerov spa vum  amboj D -n, drancic yuraqanyurum f - verlu vum  Hartogsi arqi: Bayc asti{ anayin arqi verlu um miakn , usti ayd bolor stacva ar{ qer hamnknum en irar het: U -eri proyekcianer lracnum en amboj D′ -, uremn, bolor gm -er holomorf en D′ -um: Nkatenq, or astianayin arqi andamner xmbavorelov sta{ num enq Hartogsi arq, ori zugamituyan tiruy karo  linel aveli layn: Berenq hamapatasxan rinak:

 r i n a k 1.5. Het yal ∞ ∑ |k|=0

z1k1 z2k2

Glux Լ. Holomorf funkcianer

astianayin arqi zugamituyan tiruy { } z ∈ C2 : |z1 | < 1, |z2 < 1

miavor bidiskn : Xmbavorelov nra andamnern st anneri, stanum enq Hartogsi arq` ∞ ∑ ∞ ∞ ∑ ∑ ( z1k1 )z2m = m=0 k1 =0

m=0

ori zugamituyan tiruy` haytoren, aveli layn :

{

z2 -i

asti{

z2m , 1 − z1

} z ∈ C2 : z1 ̸= 1, |z2 | < 1 ,

akn{

3. Hartogs-Lorani arqer: it aynpes, inpes astia{ nayin arqic stacvum en Hartogsi arqer, Lorani arqic l stacvum en ∞ ∑

gm (e z )(zn − an )m ,

m=−∞

ayspes kova H a r t o g s - L o r a n i  a r q e r: Aydpisi arqeri zugamituyan tiruy anvanvum  Hartogsi ki{ salriv tiruy bnuagrvum  nranov, or kam i hatvum hamaa uyan zn = 0 haruyan het, kam l amen mi z 0 keti het mekte parunakum  bolor ayn z keter, oronc hamar e ze ∈ D |zn | 6 |zn0 | : 4.

arqer st hamase bazmandamneri: pk (z) bazman{

dam kovum  k -rd astiani hamase , ee pk (ζz) = ζpk (z) bolor z ∈ Cn ζ ∈ C hamar: V a r  u  y u n 1.2. Apacucel, or trva payman ha{ mareq , or pk (z) -i bolor andamneri astian havasar lini k -i:

§ 7.

Ayl arqer

Ditarkenq  a r q  s t h a m a s e bazmandamneri, kam, inpes erbemn anvanum en a n k y u n a g a y i n  a r q: Aydpisi arq kareli  stanal astianayinic, veraxmbavorelov nra andamner` ∞ ∑

am z m =

|m|=0

∞ ∑ k=0

 

∑



am zm  =

|m|=k

∞ ∑

pk (z) :

k=0

Ayd veraxmbavorum karo  ndlaynel arqi zugamituyan tiruy, qani or nranq zugamitum en aveli layn dasi tiruy{ nerum` rjana : Berenq hamapatasxan rinak:  r i n a k 1.6. Ditarkenq het yal arq` ∞ ∑ (k1 + k2 )! k1 k2 z z : k1 ! k2 ! 1 2

(1.36)

|k|=0

Qani or astianayin arqi zugamituyun haskacvum  ba{ carak imastov, apa ays arqi zugamituyan tiruy i

oxvi, ee nra andamner veraxmbavorenq het yal  ov` ∞ ∑ (k1 + k2 )! |z1 |k1 |z2 |k2 = k1 ! k2 ! |k|=0   ∞ ∞ ∑ ∑ ∑ m!  |z1 |k1 |z1 |m−k1  = (|z1 | + |z2 |)m , = k1 ! (m − k1 )! m=0

m=0

k1 +k2 =m

ori hamar zugamituyan tiruyn  D = {z : |z1 | + |z2 | < 1} : Myus komic, nuyn  ov veraxmbavorelov (1.36)-i andamner, stanum enq st hamase bazmandamneri ∞ ∞ ∑ ∑ (k1 + k2 )! k1 k2 z1 z2 = (z1 + z2 )m k1 ! k2 !

|k|=0

m=0

arq, ori hamar zugamituyan tiruy D′ = {z : bazmuyunn , orn aveli layn , qan D -n:

|z1 + z2 | < 1}

Glux Լ. Holomorf funkcianer

st hamase bazmandamneri arqeri zugamituyan ti{ ruyner rjana lriv tiruynern en:

 e o r e m 1.27. Ee f

funkcian holomorf  D rjana lriv tiruyum, apa na verlu vum  D -i nersum bacarak havasaraa zugamet f (z) =

∞ ∑

(1.37)

pk (z)

k=0

arqi st hamase bazmandamneri: Ayste pk bazman{ damner kazmva en 0 ketum f -i eyloryan verlu uyan ayn andamneric, oronc hamar |m| = k : A p a c u y c: Skzbnaketov

fiqsa

tanenq kompleqs ui`

orte

lz 0 = {ζ ∈ Cn : ζ = tω, t ∈ C} ,

Qani or

0 ∈ D,

z 0 ∈ D \ {0}

ω=

ketov

z0 : |z 0 |

apa` f (ζ) =

∞ ∑

cm ζ

|m|=0

m

=

∞ ∑

pk (ζ)

k=0

astianayin arq zugamitum  skzbnaketi in-or rjakayqum nra hetq lz0 ui vra het yaln ` g(t) = f (tω) =

∞ ∑ k=0

pk (tω) =

∞ ∑

pk (ω)tk :

k=0

Qani or f ∈ O(D) , apa φ -n holomorf  {t ∈ C : |t| < R(ω)} rjanum, orte R(ω) = sup R grit verin ezr vercva  st bolor ayn R -eri, oronc hamar bavararvum  {t ∈ C : |t| 6 R} ⊂ D

§ 8.

Holomorf artapatkerumner

payman:

rjana lriv tiruyi sahmanumic het um , or , uremn, (1.37) arq zugamet  z 0 -um: Aym enadrenq M b D : ntrenq r(ω) funkcia, 0 < r(ω) < < R(ω) , q ∈ (0, 1) iv aynpes, or erb z ∈ M , apa |z| 6 qr(ω) , orte ω = z/|z| : Owremn bolor z ∈ M keteri hamar

|z 0 | < R(ω)

|pk (z)| = |pk (ω)||z|k 6 |pk (ω)|rk (ω)q k :

(1.38)

Hamaayn eyloryan verlu uyan gor akicneri hamar integra{ layin bana i pk (ω) =

2πi

∫

f (tω) dt : tk+1

(1.39)

|t|=r(ω)

Qani or

{ζ ∈ Cn : ζ = tω, |t| = r(ω)} b D,

apa ayd bazmuyan vra f - sahmana ak ` |f (tω)| 6 C : Ayste{ ic (1.39)-ic stanum enq Koii anhavasaruyunner eylori C arqi gor akicneri hamar` |pk (ω)| 6 k : Teadrelov dranq r (ω) (1.38)-i mej, stanum enq` |pk (z)| 6 Cq k ,

z∈M :

Qani or q ∈ (0, 1) , apa aysteic het um  (1.37) arqi hava{ saraa zugamituyun M -i vra:

§

8. Holomorf artapatkerumner

1. Gaa ar holomorf artapatkerumneri masin: Dicuq

D ⊂ Cn tiruyum orova  f = (f1 , . . . , fm ) vektor-funk{ cia: f - kovum  holomorf artapatkerum, ee nra bolor fk

Glux Լ. Holomorf funkcianer

komponentner holomorf en D -um: Masnavorapes, erb D ⊂ C , apa f - kovum  holomorf kor: Artapatkerman hamar z 0 (ket)kovum  o kritikakan, ∂fk matric ayd ketum uni ee hamapatasxan Yakobii ∂zi maqsimal ang: Parzvum , or holomorf artapatkerumneri hamar marit  m a q s i m u m i s k z b u n q : Orpeszi ayd skzbunq  akerpvi bavakanaa ndhanur depqi hamar, nermu enq C -hamase normi haskacuyun:

S a h m a n u m 1.9.

artapatkerum kovum  C -hamase norm, ee bavararvum en het yal paymanner` ∥ · ∥ : Cn 7→ R+

1)

∥z + w∥ 6 ∥z∥ + ∥w∥

2)

∥λz∥ = |λ| · ∥z∥

3)

∥z∥ = 0

bolor

bolor

z, w ∈ Cn

λ∈C

z ∈ Cn

hamar, hamar,

miayn miayn ayn depqum, erb

z = 0:

Menq himnakanum gor enq unenalu vklidyan polidiskayin normeri het, oronq C -hamase normi parzaguyn rinakner en:

 e o r e m 1.28 (maqsimumi skzbunq). Dicuq D ⊂ Cn , f -

holomorforen artapatkerum  D -n Cm -i mej ∥·∥ nana{ kum  in-or C -hamase norm Cm -um: Ee ∥f (z)∥ - hasnum  ir maqsimumin a ∈ D ketum, apa` artapatkerman fk komponentner g oren kaxyal en D -um,

(a) f

(b) ∥f (z)∥ -

nuynabar hastatun  D -um:

A p a c u y c: Dicuq b = f (a) B = {w ∈ Cm : ∥w∥ < ∥b∥}

§ 8.

Holomorf artapatkerumner

gund  ditarkvo normi nkatmamb: Normi ndhanur hatku{ yunneric het um , or B -n bac u ucik bazmuyun : Erb ∥b∥ = 0 , eoremi pndum aknhayt , enadrenq, ∥b∥ > 0 : Qani or b ∈ ∂B , apa norhiv B -i u ucikuyan, goyuyun uni ayd ketum B -in henman hiperharuyun, ori havasarun kgrenq het yal tesqov` Re l(w) = β, (1.40) orte

l(w) =

m ∑

ak w k=1 = el◦f (z)

β = Re l(b) :

k

Aym ditarkenq

D -um

holomorf F (z) funkcian: Mi komic |F (z)| = eRe l◦f (z) 6 6 eβ , myus komic` |F (a)| = β : st maqsimumi skzbunqi skalyar funkcianeri hamar (tes eorem 1.21-) l ◦ f (z) ≡ const D -um: Da nanakum , or f artapatkerman fk komponentner ba{ vararum en m ∑

ak fk (z) ≡ const

k=1

a nuyan, aysinqn nranq g oren kaxva en D -um: Baci dra{ nic, bolor z ∈ D hamar f - patkanum  (1.40) henman hiper{ haruyan ∂B ezri hatman, aysinqn ∥f (z)∥ = ∥f (b)∥ bolor z ∈ D keteri hamar: Maqsimumi skzbunqic het um  hnaravor ndhanracumneric mek:

varci lemmayi

 e o r e m 1.29.

Dicuq B1 ⊂ Cn B2 ⊂ Cm miavor gnder en hamapatasxanabar ∥ · ∥1 ∥ · ∥2 C -hamase

normerov, f : B1 7→ B2 holomorf artapatkerum aynpi{ sin , or f (0) = 0 : Ayd depqum` ∥f (z)∥2 6 ∥z∥1

bolor z ∈ B1 hamar:

(1.41)

Glux Լ. Holomorf funkcianer

A p a c u y c: Koordinatneri skzbnaketov or  z 0 ∈ ∂B1

ketov tanenq kompleqs ui` l(ζ) = ζz0 , ζ ∈ C : Aknhayt , or nra hatman B1 -i het ζ -i haruyan vra hamapatasxanum  U = {ζ : |ζ| < 1} rjan: Ditarkenq g(ζ) =

f (ζz 0 ) : U 7→ Cm ζ

holomorf kor, fiqsenq cankaca r ∈ (0, 1) : Aynuhet , ha{ maayn eorem 1.21-i, ∥g(ζ)∥2 6 1/r , ee {|ζ| 6 1} : Ancnelov ays anhavasaruyan mej sahmani, erb r → 1 , kstananq ∥g(ζ)∥2 6 1 , aysinqn ∥f (ζz 0 )∥2 6 ζ kamayakan ζ ∈ U ketum: Aym enadrenq z ̸= 0 kamayakan ket  B1 -ic: Ayd depqum z 0 = z/∥z∥1 ∈ ∂B1 teadrelov verjin anhavasaruyan mej ζ = ∥z∥1 , stanum enq (1.41): 2. Biholomorf artapatkerumner: Erb m vum  biholomorf artapatkerman gaa ar:

= n,

sahman{

S a h m a n u m 1.10. Dicuq D ⊂ Cn tiruyum orova f =

holomorf artapatkerum bavararum  het yal lracuci paymannerin`

= (f1 , . . . , fn ) : D 7→ Cn 1. 2.

na oxmiareq  hakadar s holomorf , nra bolor keter o kritikakan en, aysinqn, nra yakobia{ n` ∂f ∂(f1 , . . . , fn ) = ̸= 0 ∂z ∂(z1 , . . . , zn )

bolor

z∈D

keterum:

Ayd depqum f - kovum  biholomorf artapatkerum: Parzvum , or 2 -rd payman het um  1 -ic, bayc menq dra apacuyci vra kang enq a nelu:

§ 8.

Holomorf artapatkerumner

n = 1 depqum biholomorfuyun parzapes hamnknum  konformuyan het: n > 1 depqum da arden aydpes . rinak, {

w1 = z 1 w2 = 2z2

artapatkerum Myus komic,

C2 -ic C2 -i

vra biholomorf , bayc konform :

z 7→

z |z|2

konform artapatkerum o holomorf  o l hakaholomorf:

L e m m a 1.3. Dicuq fk = uk + ivk , k = 1, . . . , n funkcianer

holomorf en: Tei uni

∂(u1 , v1 . . . , un , vn ) ∂(f1 , . . . , fn ) = ∂(x1 , y1 , . . . , xn , yn ) ∂(z1 , . . . , zn )

havasaruyun: A p a c u y c: Hakiruyan hamar mtcnenq nanakumner` ∂(u, v) ∂(u1 , v1 . . . , un , vn ) = , ∂(x1 , y1 , . . . , xn , yn ) ∂(x, y) ∂(f1 , . . . , fn ) ∂f = , ∂(z1 , . . . , zn ) ∂z

ditarkenq

( S=

blokayin matric, orte Dvar  tesnel, or S −1

I (n) −iI (n) I (n) iI (n)

I (n) - n -rd

=

(

)

kargi miavor matric :

I (n) I (n) iI (n) −iI (n)

)

Glux Լ. Holomorf funkcianer

nra hakadarn : Aynuhet ` ∂(u, v) 1 I (n) −iI (n) ∂(u, v) I (n) I (n) · = = · ∂(x, y) 4 I (n) iI (n) ∂(x, y) iI (n) −iI (n) ∂u ∂u −i ∂x ∂y = 4 ∂u ∂u +i ∂x ∂y

∂v ∂v −i ∂x ∂y ∂v ∂v +i ∂x ∂y

·

I (n) I (n) = (n) iI −iI (n)

( ) ∂v ∂u ∂u ∂v −i +i −i ∂y ∂x ∂y 1 ∂x = ) ( 4 ∂u ∂u ∂v ∂v +i +i +i ∂x ∂y ∂x ∂y ∂ (u + iv) ∂ (u + iv) −i ∂x ∂y = 4 ∂ (u + iv) ∂ (u + iv) +i ∂x ∂y ∂f = ∂z ∂f ∂ z̄

∂f ∂ z̄ = ∂f ∂f ∂z ∂z

,

∂u ∂u −i −i ∂x ∂y ∂u ∂u +i −i ∂x ∂y

( (

∂v ∂v −i ∂x ∂y ∂v ∂v +i ∂x ∂y

∂ (u − iv) ∂ (u − iv) −i ∂x ∂y ∂ (u − iv) ∂ (u − iv) +i ∂x ∂y

orovhet

) ) =

=

∂f ∂f = =0: ∂ z̄ ∂ z̄

H e t a n q 1.2.

Ee D ⊂ Cn tiruyum orova f = = (f1 , . . . , fn ) : D 7→ Cn holomorf artapatkerum uni zro{ ∂f yic tarber yakobian, apa na lokal homeomorf : ∂z

A p a c u y c: Iroq, ee f - ditarkenq orpes R2n -um oro{

va artapatkerum, apa st lemma 1.3-i nra irakan yako{ bian zro i da num: Hamaayn irakan analizic haytni anba{ cahayt funkciayi masin eoremi, f - yuraqanyur keti rja{ kayqum oxmiareq  uni anndhat hakadar` da l henc nanakum , or na lokal homeomorf :

§ 8.

Holomorf artapatkerumner

Aym nkatenq, or lokal oxmiareq holomorf artapatke{ rum karo  linel oxmiareq global imastov, aysinqn, am{ boj tiruyum: Bayc ee na lini global oxmiareq, apa klini na biholomorf: Dran  nvirva hajord eorem:  e o r e m 1.30. Amen mi holomorf oxmiareq arta{ patkerum biholomorf :

A p a c u y c: st paymani, z = z(w) hakadar artapat{

kerum miareq  mnum  miayn apacucel nra holomorfuyun: Inpes irakan analizum, marit  ∂w ∂z · =1 ∂z ∂w ∂w a nuyun, orteic het um , or ∂z

yic: Havenq wk = wk (z(w)), mal a ancyalner`

yakobian tarber  zro{

k = 1, . . . , n

bard funkcianeri for{

) n ( ∑ ∂wk ∂zm ∂wk ∂ z̄m ∂wk = + : ∂ w̄j ∂zm ∂ w̄j ∂ z̄m ∂ w̄j m=1

Aj masum aknhaytoren

∂wk ∂wk = =0 ∂ w̄j ∂ z̄m

uremn,

n ∑ ∂wk ∂zm =0: ∂zm ∂ w̄j

m=1

Stacanq n g ayin havasarumneri hamakarg, ori oroi ∂w ̸= 0 : Owremn na uni miak lu um` ∂z

∂zm = 0, ∂ w̄j

Da nanakum , or morf :

z = z(w)

m, j = 1, . . . , n :

hakadar artapatkerum holo{

Glux Լ. Holomorf funkcianer

3. Avtomorfizmner: Biholomorf artapatkerum kovum  tiruyi avtomorfizm, ee na artapatkerum  D -n inqn ir vra: Kompoziciayi gor ouyan nkatmamb avtomorfizmner kazmum en xumb, or nanakvum  Aut D : Mer motaka npatak klini nkaragrel gndi polidiski avtomorfizmner: D

M i a v o r g n d i a v t o m o r f i z m n e r  : Hiecnenq, or miavor rjanin patkano amen mi α vin hamapatasxa{ num  rjani avtomorfizm, or tea oxum  α 0 keter, α−z : Parzvum , or nuyn pndum it  na ayn ` φα (z) = 1 − ᾱz n C -um miavor gndi depqum: Dicuq B n = {z ∈ Cn : |z| < 1} : Nax ka ucenq avtomorfizm, or tvyal fiqsa a ∈ B n ket artapatkerum  z = 0 kentroni: Nanakenq pa (z) -ov z keti proyekcian 0 a keterov ancno la kompleqs ui vra qa (z) -ov nra proyekcian la -i rogonal lracman vra: Het  tesnel, or pa (z) =

⟨z, a⟩ a, |a|2

orte

⟨z, a⟩ =

n ∑

zk āk ,

k=1

isk qa (z) = z − pa (z) : a = 0 depqum hamarum enq p0 (z) Apacucenq, or oroneli avtomorfizm het yal tesqi ` φa : w =

orte

α=

√

1 − |a|2 :

a − pa (z) − αqa (z) , 1 − ⟨z, a⟩

≡ 0:

(1.42)

Ays artapatkerum orova 

Ω = {z ∈ Cn : ⟨z, a⟩ ̸= 1}

tiruyum, nd orum Ω ⊃ B n , orovhet st varci anhavasa{ ruyan |⟨z, a⟩| 6 |a| |z| |a| < 1 : Qani or pa (a) = a qa (a) = 0 , apa φa (a) = 0 :

§ 8.

Holomorf artapatkerumner

Hark ea depqum katarelov Cn tara uyan unitar  a{

oxuyun, karo enq enadrel, or a = (e0, an ) : Ayd depqum pa (z) = (e 0, zn ) , qn (z) = (e z , 0) (1.23)- ndunum  w e = −α

tesq: Havelov

|w|2 ,

|w|2 =

ze , 1 − ān zn

wn =

an − zn 1 − ᾱn zn

(1.43)

kstananq`

|z|2 + |a|2 (1 − |e z |2 ) − 2Re(ān zn ) , 1 + |ān zn |2 − 2Re(ān zn )

orteic er um , or ee |z| < 1 , apa |w| < 1 , isk ee |z| = apa |w| = 1 : Ayspisov, φa -n artapatkerum  B n -n inqn ir vra: gtvelov (1.43)-ic, kareli  hamozvel, or φa ◦ φa (z) ≡ z : Dranic nax het um , or φ−1 heto or φa -n oxmiareq a = φa , n n  ov  artapatkerum B -n B -i vra, aysinqn, φa -n biholo{ morf : Parz , or ee U -n unitar perator  Cn -um, apa U ◦φa -n noric patkanum  Aut B n -in: Apacucenq, or drancov spa vum en gndi bolor avtomorfizmner:

= 1,

 e o r e m 1.31. Cankaca f ∈ Aut B n -i hamar goyuyun

unen a ∈ B n ket or f = U ◦ φa :

U

unitar perator Cn -um aynpisin,

A p a c u y c: Dicuq a -n 0 keti naxapatkern ` f (a) = = 0 : Ayd depqum g = f ◦ φ−1 artapatkerum inpes nra a −1 g hakadar, biholomorf en gndum anar en onum nra

kentron: Drancic yuraqanyuri nkatmamb kira elov varci lemman vklidyan normov, kstananq |g(z)| 6 |z| |g−1 (w)| 6 |w| amenureq B n -um, orteic het um ` |g(z)| ≡ |z|,

z ∈ Bn :

(1.44)

Glux Լ. Holomorf funkcianer Aym fiqsenq

rjanum ditarkenq G(ζ) = vektor-funkcian: Qani or g(0) = 0 , G -n holomorf ζ  miavor rjanum: 1.44-ic het um  |G(ζ)| ≡ 1 : Havi a nelov gndi xist u ucikuyun, maqsimumi skzbunqic ezrakacnum enq, or G -n hastatun ` G(ζ) = c(z 0 ) : Ayspisov, g(ζz 0 ) = c(z 0 )ζ , in nanakum , or g -n g ayin  n B -i amen mi z = ζz 0 kompleqs haruyan hatman vra g(ζz 0 )

z 0 ∈ ∂B n

{|ζ| < 1} ⊂ C

g(λz) = λg(z),

λ ∈ C, |λ| < 1 :

(1.45)

Verlu elov g -i koordinatner arqi st hamase bazmandam{ neri, stanum enq g(z) =

∞ ∑

Pk (z)

k=0

verlu uyun, orte Pk -er hamase vektor-bazmandamner en: Havi a nelov (1.45)-, aysteic stacvum , or cankaca λ ∈ C ( |λ| < 1 ) vi hamar` g(λz) =

∞ ∑ k=0

k

λ Pk (z) = λ

∞ ∑

Pk (z) :

k=0

Ditarkelov stacva arqer orpes astianayin st ζ -i, mia{ kuyan eoremic ezrakacnum enq, or bolor Pk -er, baci P1 -ic, nuynabar havasar en zroyi: Ayspisov, g -n g ayin  , inpes vkayum  (1.44)-, na unitar ` g = f ◦ φ−1 = U : Ayspisov, a f = U ◦ φa : Isk aym havenq Aut B n xmbi ankax parametreri qanak, oroncic na kaxva : (1.42)-ic er um , or φa -n orovum  a ketov, aysinqn, kaxva  2n irakan parametreric: Myus ko{ mic, unitar U  a oxuyan hamapatasxanum  unitar A = (akj ) matric, ori tarrer bavararum en n2 irakan

§ 8.

Holomorf artapatkerumner

a nuyunnerin: Iskapes, unitar linelu payman artahayt{ ∑ vum  ni=1 aki āji = δkj havasarumnerov, orte δkj -n Kronekeri simvoln : Ayd havasarumneri qanak havasar  n2 : Drancic n hat (erb k = j ) irakan en, isk mnaca ner` zuyg a zuyg kompleqs hamalu en, aynpes or mnum en (n2 − n)/2 kompleqs, aysinqn n2 − n irakan paymanner: Ayspisov, Aut B n xumb kaxva  n2 + n irakan parametreric: Nkatenq Aut B n xmbi mi kar or hatkuyun. cankaca a ∈ ∈ Bn b ∈ B n keteri hamar goyuyun uni φ avtomorfizm, ori hamar φ(a) = b : Iroq, orpes φ kareli  vercnel φ = φb ◦ φa artapatkerum: P o l i d i s k i a v t o m o r f i z m n e r  : Nax nkatenq, or wk = eiθk

zk − ak , 1 − āk zk

k = 1, . . . , n

(1.46)

kotorakag ayin  a oxuyunnern aknhaytoren patkanum en Aut U n -in: Katarelov koordinatneri wk 7→ wσ(k) tea oxu{ yun, kstananq nor avtomorfizmner: Parzvum , or drancov spa vum en polidiski bolor avtomorfizmner:  e o r e m 1.32. Aut U n xumb kazmva  zk 7→ eiθσ(k)

zσ(k) − aσ(k) , 1 − āσ(k) zσ(k)

k = 1, . . . , n

(1.47)

tesqi bolor  a oxuyunneric, orte σ -n (1, . . . , n) baz{ muyan kamayakan tea oxuyun : A p a c u y c: Dicuq f ∈ Aut U n , a = f (0)

g -n (1.46) ba{ na ov orovo artapatkerum : Ayd depqum F = g ◦ f - biho{ lomorforen artapatkerum  U n - ir vra aynpes, or F (0) = 0 : Kira elov F -i F −1 -i nkatmamb varci lemman polidiskayin

normov, kstananq, inpes naxord eoremum, or ∥F (z)∥ = ∥z∥

bolor

z ∈ Un :

(1.48)

Glux Լ. Holomorf funkcianer

Aym ditarkenq F artapatkerman or  Fm koordinat: Qani or ∥F (z)∥ = 16i6n max |Fi (z)| F (U n ) = U n , apa U n -um goyuyun uni bac enabazmuyun, orte ∥F (z)∥ = |Fk (z)| : Ayd enabazmuyan naxapatkerum in-or j -i hamar kunenanq ∥z∥ = |zj | : Kira elov Fk funkciayi nkatmamb varci miaa

lemman, havi a nelov (1.48)-ic het o |Fk (z)| = |zj | havasa{ ruyun, kstananq Fk (z) = eiθ(z) zj : Ayste θ -n karo  kaxva

linel mnaca zi koordinatneric, bayc qani or eiθ(z) - holomorf  hastatun  ir modulov, apa na hastatun ` θ(z) = θj : Ayspisov, Fk (z) = eiθj zj bac enabazmuyan vra, uremn am{ boj U n -um st miakuyan eoremi: Ev verjapes, F -i oxmiareq lineluc het um , or j = = j(k) -n handisanum  (1, . . . , n) indeqsneri tea oxuyun: Da nanakum , or f = g−1 ◦ F artapatkerum (1.47) tes{ qi : eoremic het um , or Aut U n xumb bnakan  ov trohvum  (1, . . . , n) bazmuyan tea oxuyunneric a ajaca n! hat enaxmberi, oroncic yuraqanyur kaxva  3n irakan pa{ rametreric. n hat irakan θk nuynqan kompleqs ak veric: Stacvec, or Aut U n xumb kaxva  3n irakan paramet{ reric: Inpes tesnum enq, B n -um U n -um avtomorfizmner kaxva en tarber qanakov parametreric. gndi depqum n2 + + 2n , isk polidiski` 3n : Ownenalov Aut B n Aut U n xmberi nkaragruyun, karo enq apacucel het yal hetaqrqir ast, or nkatel r Puan{ karen de 1907 vakanin: Bn

 e o r e m 1.33 (Puankare). gndi

Un

n > 1 depqum goyuyun uni polidiski biholomorf artapatkerum:

A p a c u y c: Enadrenq, goyuyun uni f : B n 7→ U n biho{

§ 8.

Holomorf artapatkerumner

lomorf artapatkerum: Ayd depqum goyuyun kunenar f ∗ : Aut B n 7→ Aut U n

hamapatasxan xmberi izomorfizm f ∗ : φ 7→ f ◦ φ ◦ f −1 ,

φ ∈ Aut B n

bana ov: Bayc da hnaravor , orovhet , inpes tesanq, ayd xmber kaxva en tarber qanakov parametreric. Berenq mi ayl apacuyc: Dicuq a = f (0) , orte f - ver nva enadrveliq artapatkerumn  vercnenq g ∈ Aut U n aynpisin, or g(a) = 0 : Ayd depqum F = g ◦ f -n biholomorfo{ ren artapatkerum  B n - U n -i vra aynpes, or F (0) = 0 : Kira elov inpes naxord eoremnerum F -i F −1 -i nkatmamb varci lemman, kstananq, or ∥F (z)∥ = |z| bolor z ∈ B n keteri hamar: Aysteic het um , or {z : |z| = 1/2} vklidyan gndolort F - artapatkerum  o oork {z : ∥z∥ = 1/2} maker uyi, in hnaravor , orovhet F - difeomorfizm : Ayspisov, parzvec, or imani eorem haruyan vra mia{ kap tiruyneri konform hamarequyan masin Cn tara u{ yunum n > 1 depqum marit : Inpes miaa depqum, gndi u polidiski avtomorfizmner kotorakag ayin artapatkerumner en: vum , e nmanuyamb amboj Cn tara uyan avtomorfizmner petq  linen g ayin, bayc da aydpes : rinak, f : (z1 , z2 ) 7→ (z1 + φ(z2 ), z2 )

tesqi artapatkerum, orte φ -n mek o oxakani kamaya{ kan anboj funkcia , C2 tara uyan avtomorfizm : Iroq, inq, nra f −1 : (w1 , w2 ) 7→ (w1 − φ(w2 ), w2 )

Glux Լ. Holomorf funkcianer

hakadar holomorf en C2 -um: Fatui komic ka ucvel  f : C2 (7→ )C2 biholomorf arta{ patkerman rinak, ori depqum C2 \f C2 ndunvo areqneri bazmuyun parunakum  o datark bac bazmuyun: Ayd ri{ nak vkayum , or Pikari eorem ir uaki  akerpumov bazmaa depqum it : Kan ayd eoremi ndhanracumner tarber  akerpumnerov, bayc dranc vra kang enq a ni:

§

9. Gaa ar meromorf funkciayi masin

1. Meromorf funkciayi sahmanum: f funkcian kovum

 meromorf

G

tiruyum, ee

1. holomorf  amenureq G -um, baci in-or

P

bazmuyunic,

2. analitikoren i arunakvum P -i o mi ket, 3. kamayakan z 0 ∈ P keti hamar goyuyun uni U ∩ rjakayq U -um holomorf ψ ̸≡ 0 funkcia aynpisiq, or G (U \ P ) bazmuyan vra holomorf φ = f ψ funkcian analitikoren arunakvum  U -i mej: ∩

Parz , or ψ(z 0 ) = 0 yuraqanyur z 0 ∈ P U ketum. haka ak depqum f ψ funkciayi het mekte f - s karunakver z 0 keti or  rjakayq: Enadrenq, kamayakan z 0 ∈ P keti hamar φ ψ funkcianer unen z 0 -um holomorf ndhanur artadriner, oronq zro en da num ayd ketum. haka ak depqum kareli  φ -n ψ -n kratel aydpisi artadrineri vra: Ayspisov, P -n anali{ tik bazmuyun , orovhet iren patkano kamayakan z 0 keti rjakayqum na orovum  P = {z ∈ U : ψ(z) = 0}

§ 9.

Gaa ar meromorf funkciayi masin

paymanov: P bazmuyun kovum  f funkciayi b e ayin baz{

muyun:

B e ayin bazmuyan tarber keterum f -i varq karo  linel tarber: z 0 ∈ P ket kovum  b e , ee φ = f ψ funkcian tarber  zroyic ayd ketum, kovum  anorouyan ket, ee φ φ = 0 : B e in motenalis f = funkcian gtum  anverjuya{ ψ n, isk anorouyan keti rjakayqum na ndunum  kamayakan areq: Iskapes, z 0 -n parunako {z ∈ U : φ(z) − w0 ψ(z) = 0}

analitik bazmuyan vra

f ≡ w0 : z 1.7 f (z) = 2 funkcian meromorf  C2 -um, z1 bazmuyun {z1 = 0} kompleqs uin : Ayd ui b e ner en, baci {z1 = 0, z2 = 0} ketic, or

rinak

nra b e ayin bolor keter anorouyan ket :

.

2. Kuzeni himnaxndir: Inpes haytni , mek o oxakani

depqum hnaravor  ka ucel meromorf funkcia, orn uni tvyal b e nern u glxavor maser:  akerpenq ayd xndir het yal  ov: Dicuq G ⊂ C tiruyum trva  G -um sahmanayin ket uneco ak keteri gk (z) =

nk ∑ m=1

(k)

cm (z − ak )m

funkcianeri hajordakanuyun: Ditarkenq G tiruyi a { kuy Uα ⊂ G enatiruynerov, oroncic yuraqanyur paru{ nakum  verjavor vov ak keter nanakenq fα -ov gk -eri gumarn st ak ∈ Uα keteri: Ee Uα -n ak keter i parunakum, apa khamarenq fα ≡ 0 : Bolor fα -ner meromorf en, nd orum, ∩ ee Uα Uβ ̸= ∅ , apa ayd hatman vra fα − fβ = hαβ funkcian

Glux Լ. Holomorf funkcianer

holomorf : Anhraet  G -um ka ucel aynpisi meromorf f funkcia, or (f − fα ) tarberuyun lini holomorf Uα -um bolor α -neri hamar: st Mittag{Lefleri eoremi, ayd xndir mit uni lu um kamayakan har G tiruyi hamar: Aydpisi tesqov xndir uyl  talis ndhanracum kama{ yakan G ⊂ Cn tiruyi hamar kovum  Kuzeni a ajin himnaxndir:

Stor enq nra ∪

G

Uα Uα = G

α∈A

tiruyi {Uα }α∈A a kuy aselov haskanalu bac enabazmuyunneri aynpisi hamakarg, or amen mi p ∈ G ket patkanum  verjavor vov

Uα -nerin:

Ayd himnaxndri drva q het yaln .

Trva  G ⊂ Cn tiruyi {Uα }α∈A a kuy yuraqanyur Uα -um meromorf fα funkcia, nd orum o datark Uαβ = ∩ = Uα Uβ hatumneri vra fα − fβ = hαβ funkcianer ho{ lomorf en: Pahanjvum  G -um ka ucel aynpisi meromorf f funkcia, or (f − fα ) -n lini holomorf Uα -um bolor α -neri hamar:

Bazmaa depqum ( n > 1 ) Kuzeni a ajin himnaxndir o mit uni lu um (tes xndir 2.20):

Xndirner

Xndirner co

X n d i r 1.1. Artahaytel (1.2) havasarumneri mej masnak{ aik , a′ik

bi

ver (1.1)-i

αik

βi

veri mijocov:

X n d i r 1.2. Apacucel, or Cn tara uyan Rnx

ani haruyunner kompleqs haruyunner en:

X n d i r 1.3. Nkaragrel

tuyner

{z ∈ Cn : |z| = 1} z = a + ωζ (a, ω ∈ Cn ; ζ ∈ C) kompleqs

X n d i r 1.4. Nkaragrel {z ∈ C2 :

|z| < 1}

Rny n -a{

gndolorti ha{ uinerov:

gndi

{z ∈ C2 : |z1 | < 1, |z2 < 1}

bidiski hatumner α -neri depqum:

y2 = α

e aa haruyunnerov, tarber

X n d i r 1.5. Apacucel, or

uyan cankaca ketov hiperharuyun:

Cn -um irakan S hiperhar{ ancnum  S -in patkano kompleqs

X n d i r 1.6. Apacucel, or

g oren u ucik :

Cn -um

u ucik tiruy na

X n d i r 1.7. Berel Cn -um g oren u ucik tiruyi rinak,

or sakayn u ucik :

X n d i r 1.8. Berel Cn -um tiruyi rinak, or rjana ,

sakayn n -rjana :

Glux Լ. Holomorf funkcianer

X n d i r 1.9. Oroel het yal arqeri zugamituyan ti{

ruyner. a) b) g)

∞ ∑

(z1 z2 )k ,

k=0 ∞ ∑

(z1 z2 )k +

k=0 ∞ ∑

(z1 z22 )k +

k=0

∞ zk + zk ∑ , k k=0 ∞ ∑

(z12 z2 )k :

k=0

X n d i r 1.10. Ka ucel astianayin arq, ori zugamitu{ yan tiruyn ` a)

{z ∈ C2 : |z1 | + |z2 | < 1}

tiruy,

b)

{z ∈ C2 : |z1 |2 + |z2 |2 < 1}

gund:

X n d i r 1.11. Ka ucel astianayin arq, ori hamar zu{

gamituyan bazmuyunn `

∪ {z ∈ Cn : |z| < 1} {|z| < 2, z2 = 0} :

X n d i r 1.12. Cuyc tal, or

∞ ∑ m,n=0

amn z1m z2n

astianayin

arq, ori gor akicner kazmum en 0! 1! 2! 3!

1! 2! 3! · · · −1! −2! −3! · · · −2! 0 0 ··· −3! 0 0 ··· ··· ··· ··· ···

anverj matric, o bacarak zugamitum  (1,1) ketum ta{ ramitum  C2 tara uyan mnaca bolor keterum (hava , iharke, skzbnaket): Sa nanakum , or Abeli eorem ir sovo{ rakan  akerpumov it  bazmapatik arqeri hamar:

Xndirner

X n d i r 1.13. Berel n -harmonik funkciayi rinak, or plyuriharmonik : X n d i r 1.14. Apacucel, or D ⊂ Cn tiruyum erku angam

anndhat diferenceli u(z) funkcian plyuriharmonik  ayn miayn ayn depqum, erb ∂u diferencial   ak :

X n d i r 1.15. Apacucel, or orpeszi

tiruyum u(z) funkcian lini plyuriharmonik, anhraet  bavarar, or cankaca a ∈ D keti rjakayqum na lini holomorf funkciayi irakan mas:

X n d i r 1.16. Dicuq

D ⊂ Cn

funkcian D ⊂ Cn tiruyum erku angam anndhat diferenceli : Apacucel, or u(z) - plyu{ riharmonik  D -um ayn miayn ayn depqum, erb nra hetq ka{ mayakan l kompleqs ui vra harmonik  l ∩ D -um: u(z)

X n d i r 1.17. Dicuq u(z) funkcian analitik  st irakan

koordinatneri D ⊂ Cn tiruyum plyuriharmonik  or  V ⊂ ⊂ D gndum: Apacucel, or u(z) - plyuriharmonik  amboj D -um:

X n d i r 1.18. Dicuq f -

bavararum 

Cn -um

amboj funkcia , or

|f (z)| 6 C (1 + |z|m )

anhavasaruyan, orte C -n m - hastatun me uyunner en: Apacucel, or f - bazmandam , ori astian i gerazancum m iv:

X n d i r 1.19. Dicuq f - holomorf  E ⊂ Cn miavor poli{

diskum,

|f (z)| 6 M

f (0) = 0 :

Apacucel, or

|f (z)| 6 M ρ(z),

z ∈ E,

orte ρ(z) = max |zk | : 16k6n (Sa varci lemmayi bazmaa nmanakneric mekn :)

Glux Լ. Holomorf funkcianer

X n d i r 1.20. Dicuq M - kompleqs haruyan vra miavor rjani enabazmuyun , ori hamar 0 -n xtacman ket , isk f (z1 , z2 ) - miavor E bidiskum orova sahmana ak funkcia , or holomorf  st z1 -i, erb |z2 | < 1 , holomorf  st z2 -i, erb z1 ∈ M : Apacucel, or f - holomorf  E -um: X n d i r 1.21. Apacucel, or ee f (z) = f (z1, . . . , zn) funk{ cian bazmandam  st yuraqanyur zν -i, ν = 1, . . . , n , apa f - bazmandam : X n d i r 1.22. Cuyc tal, or

holomorf

f (z) =

funkcian anndhat 

B -um,

B = {z ∈ Cn : |z| < 1}

gndum

z13 1 − z22

bayc i nerkayacvum

f (z) = z1 φ(z1 , z2 )

tesqov, orte φ -n holomorf 

B -um

anndhat 

B -um:

X n d i r 1.23. Dicuq E -n miavor polidiskn  Cn -um

∈ O(E) ∩ C(E) :

Apacucel, or f - holomorf  yuraqanyur

f ∈

∆j,a = {z ∈ Cn : zm = am , |am | 6 1, m = 1, · · · , n, m ̸= j, |ζj | < 1}

rjanum:

X n d i r 1.24. Apacucel, or ee f funkcian holomorf  0 ∈

keti rjakayqum havasar  zroyi irakan haruyan vra, apa f ≡ 0 ayd rjakayqum: ∈ Cn

X n d i r 1.25. Apacucel, or ee

funkcian holomorf  0∈ keti rjakayqum havasar  zroyi {z ∈ C2 : z1 = z̄2 } haruyan vra, apa f ≡ 0 ayd rjakayqum: C2

f

Xndirner

X n d i r 1.26.

Cn -um ka ucel keteri hajordakanuyun, or zugamitum  E miavor polidiski kentronin miakuyan bazmuyun  O(E) dasi hamar:

X n d i r 1.27.

B = {z ∈ Cn : |z| < 1} gndi ezri korer, oroncic o mek O(B) ∩ C(B)

vra ka ucel dasi hamar oronc miavorum aydpisi bazmu{

haveli vov miakuyan bazmuyun  yun : X n d i r 1.28. B = {z ∈ Cn : |z| < 1} gndi ezri vra ka ucel O(B)∩C(B) dasi hamar ak miakuyan bazmuyun, ori g ayin a  verjavor : X n d i r 1.29. Apacucel, or E ⊂ Cn miavor polidiski henqi kamayakan o datark bac enabazmuyun O(E) ∩ C(E) dasi hamar miakuyan bazmuyun : X n d i r 1.30. Apacucel, or E ⊂ Cn miavor polidiskum ka{ mayakan n -harmonik f (z) funkciayi hamar tei uni Puasoni bazmaa f (z) = P [f ](z) bana : X n d i r 1.31. Dicuq f ∈ C(Γ) fk ∈ C(Γ) , orte Γ -n E miavor polidiski henqn , fk → f : Apacucel, or lim P [fk ](z) = P [f ](z)

k→∞

havasaraa E -i vra: X n d i r 1.32. Apacucel, or ee f ∈ C(Γ) , orte Γ -n E miavor polidiski henqn , apa nra P [f ](z) Puasoni integral anndhatoren arunakvum  E -i vra: X n d i r 1.33. Dicuq f ∈ C(Γ) , orte Γ -n E miavor polidis{ ki henqn : Apacucel, or orpeszi f - arunakvi min O(E) ∩ C(E) dasi funkcia, anhraet  bavarar, or ∫

f (ζ)ζ k dζ = 0 Γ

Glux Լ. Holomorf funkcianer bolor k = (k1 , . . . , kn ) vektorneri hamar, orte en nrancic gone mek o bacasakan :

kν -er

amboj

X n d i r 1.34. Dicuq f funkcian orova anndhat  E

miavor polidiski

∂E

ezri vra yuraqanyur

∆j,a = {z ∈ Cn : zm = am , |am | 6 1, m = 1, · · · , n, m ̸= j, |ζj | < 1}

rjanum holomorf : Apacucel, or f - arunakvum  min ∩ C(E) dasi funkcia:

O(E)∩

X n d i r 1.35. Apacucel, or kamayakan funkcia, or holo{

morf 

{ } { } D = z ∈ C2 : |z1 | < 2, |z2 | < 2 \ z ∈ C2 : |z1 | < 1, |z2 | < 1

snamej bidiskum, (g agrum trvum  D -i eynhartyan diagra{ { } m), analitikoren arunakvum  z ∈ C2 : |z1 | < 2, |z2 | < 2 bi{ diski mej: Èz2 È

Èz1 È

GLUX

HOLOMORFUYAN TIRUYNER

§

1.

10. Analitik arunakuyun

arunakuyun ezri rjakayqic. A ajin glxum menq ar{

den handipel enq mi er uyi, or yurahatuk  miayn mi qani

o oxakani holomorf funkcianerin, ayn ` ayspes kova har{ kadir analitik arunakman het: eorem 1.25-um pndvum r, or gndayin ertum amen mi holomorf funkcia analitikoren aru{ nakvum  gndi mej (nuyn polidiskayin erti hamar tes` xn{ dir 1.35-um): Stor berva eorem 3.14-ic het um , or ayd er uy uni ndhanur bnuy. holomorf funkcian i karo unenal kompakt ezakiuyunner:

 e o r e m 2.1.

Dicuq Ω -n tiruy  Cn -um, n > 1 , K -n aynpisi kompakt enabazmuyun  Ω -um, or Ω \ K -n ka{ pakcva : Ayd depqum kamayakan g ∈ O(Ω \ K) funkcia kareli  analitikoren arunakel amboj Ω -i vra: A p a c u y c: Dicuq φ ∈ C ∞ (Cn ) aynpisin , or φ ≡ 1 K

kompakt bazmuyan V rjakayqum kri: Ka ucenq het yal (0, 1) -  { f=

¯ g ∂φ

φ -n uni kompakt K0 ⊂ Ω

Ω \ K -um, Cn -i

mnaca bolor keterum:

¯ = 0 V -um K0 -ic durs, apa f - orova  amboj Qani or ∂φ Cn -um, uni C ∞ -gor akicner nra kri nka  K0 -i mej:

Glux ԼԼ. Holomorfuyan tiruyner

Dicuq Ω0 -n Cn \Ω0 -i ansahmana ak komponentn , dicuq ¯ = f havasarman ayn lu umn , or havasar  zroyi u -n ∂u Ω0 -um: st eorem 4.14-i aydpisi lu um goyuyun uni: Aynu{ het ka ucenq {

G=

u + (1 − φ)g u

Ω \ K -um V -um

funkcian: Qani or φ(z) ≡ 1 , erb z ∈ V , apa G -i sahmanum ko ekt  G ∈ C ∞ (Ω) : Cuyc tanq, or G ∈ O(Ω) : Iroq, V -um tei uni ¯ = ∂u ¯ =f =0 ∂G

havasaruyun, isk

Ω \ K -um`

¯ = ∂u ¯ − g ∂φ ¯ = f − f = 0, ∂G ¯ = 0: qani or ∂g Ev, verjapes, Ω0 ∩ (Ω \ K) bazmuyan vra φ = 0 het abar G = g : Ow qani or Ω0 ∩(Ω\K) -n o datark  kapakcva , apa G -n g -n hamnknum en amenureq vra:

u = 0, Ω\K -n Ω \ K -i

Ayspisov, amen mi funkcia, or holomorf  tiruyi ezri rjakayqum, analitikoren arunakvum  amboj tiruyi mej: Ays pndum kareli  ueacnel, pahanjelov, or arunakvo funkcian orova lini miayn ezri vra in-or imastov lini holomorf, bavarari ayspes kova Koi{ imani oa o hava{ sarumnerin: Ancnenq grit sahmanumnerin: Cn

2. Koi{ imani oa o perator. Dicuq D -n tiruy 

-um

ρ -n

irakan

C 2 -funkcia  D -um: Nanakenq ( ) ∂ρ ∂ρ , ζ∈D: N (ζ) = ∂ ζ̄1 ∂ ζ̄n

§ 10.

Dicuq

ΩbD

Analitik arunakuyun

tiruyi hamar ρ -n oroi funkcia , aysinqn

Ω = {z ∈ D : ρ(z) < 0} ,

Aynuhet , dicuq cia 

N (ζ) ̸= 0,

a : D 7→ Cn \ {0} L=

n ∑

āk (z)

k=1

erb

ζ ∈ ∂Ω :

anndhat vektor-funk{

∂ ∂ z̄k

hamapatasxan diferencial peratorn : Aknhayt , or Lf = = 0 bolor f ∈ O(D) funkcianeri hamar, ayd pata ov L - kovum  Koi{ imani perator: Kira elov L - oroi ρ funk{ ciayi nkatmamb, kstananq Lρ(z) = ⟨N (z), a(z)⟩ ,

z∈D:

S a h m a n u m 2.1. L - kovum  oa o perator, ee Lρ(ζ) = 0,

kam, or nuynn ,

a ⊥ N ∂G -i

ζ ∈ ∂G,

bolor keterum:

Dicuq u1 , u2 ∈ C 1 (D) u1 (ζ) = u2 (ζ) bolor ζ ∈ ∂G hamar: Nanakelov u = u1 − u2 kunenanq u = 0 ∂G -i vra uremn, grad u -n hamematakan  grad ρ -in ∂G -i keterum: Ayspisov, goyuyun uni funkcia h : ∂G 7→ C aynpisin, or ∂ρ(ζ) ∂u(ζ) = h(ζ) , ∂ ζ̄k ∂ ζ̄k

k = 1, . . . , n,

ζ ∈ ∂G :

Hielov L -i sahmanum, aysteic stanum enq Lu(ζ) = h(ζ)Lρ(ζ),

Het abar ∈ ∂G -i

C 1 (D)

vra:

ζ ∈ ∂G :

Lu1 = Lu2 ∂G -i vra: Ayl ba erov asa , ee f ∈ ζ ∈ ∂G , apa Lf (ζ) -n kaxva  miayn f -i hetqic

Glux ԼԼ. Holomorfuyan tiruyner

Owremn menq karo enq ditarkel L - orpes perator, or gor um  C 1 (∂G) -i vra. ee f ∈ C 1 (G) , apa Lf - kaxva  f -i C 1 -arunakuyunic oric menq gtvum enq a ancyalner havelu hamar: P n d u m 2.1. Dicuq u ∈ C 1(D) : Ayd depqum het yal paymanner hamareq en ¯ ∧ ∂ρ ¯ = 0 ∂G -i vra, a) ∂u

b) Lu = 0 Koi{ imani bolor L peratorneri hamar, oronq oa o en ∂G -i hamar: A p a c u y c: Het  tesnel, or ) ∑ ( ∂ρ ∂u ∂ρ ∂u ¯ ∧ ∂ρ ¯ = ∂u − dz̄j ∧ dz̄k : ∂ z̄k ∂ z̄j ∂ z̄j ∂ z̄k j<k

Nermu enq Ljk =

∂ρ ∂ ∂ρ ∂ − , ∂ z̄k ∂ z̄j ∂ z̄j ∂ z̄k

16j<k6n

peratorner: Inpes er um , Ljk -er oa um en ∂G -in bavararum en a) paymanin ayn miayn ayn depqum, erb Ljk u(ζ) = 0, j < k, ζ ∈ ∂G : (2.1) Ver um trva sahmanumneri terminnerov Ljk -eri hamapa{ tasxan vektorner klinen a = ajk = ∂ρ(ζ)

∂ρ(z) ∂ρ(z) ej − ek : ∂zk ∂zj

Ee ζ ∈ ∂G , apa ̸= 0 or  m -i hamar: Ayd depqum ajm ∂ζm amk ( 1 6 j < m < k 6 n ) n−1 hat vektorner g oren ankax en: Owremn, nranq a ajacnum en ζ ketum ∂G -in kompleqs oa o tara uyun , inpes het um  (2.1)-ic, Lu = 0 Koi{ imani ∂G -in oa o kamayakan L peratori hamar:

§ 10.

Analitik arunakuyun

 r i n a k 2.1. Dicuq D = Cn , ρ(z) = |z|2 −1 , het abar, G -n

miavor gund : Ver um sahmanva

depqum unen Ljk = ζk

Ljk

peratorner tvyal

∂ ∂ − ζj : ∂ ζ̄j ∂ ζ̄k

tesq: Kasenq, or S = ∂B -i vra tvyal u ∈ C 1 (S) funkcian bavararum  Koi{ imani oa o havasarumnerin, ee ζk

∂u ∂u = ζj , ∂ ζ̄j ∂ ζ̄k

1 6 j < k 6 n,

ζ∈S:

Nenq, or n = 2 depqum ays hamakarg bervum  mek havasar{ man: Hajord eorem handisanum  eorem 2.1-i ueacum ayn imastov, or arunakvo funkcian orova  o e ezri rja{ kayqum, ayl miayn ezri vra: Nra apacuyc katarvum  nman meodov, in nva eoremum, ∂¯ -xndri lu man kira uyamb2 :  e o r e m 2.2 (Boxner). Dicuq n > 1 , Ω -n sahmana ak

tiruy  Cn -um C 4 -ezrov Cn \ Ω -n kapakcva : Ayd dep{ qum kamayakan u funkcia C 4 (∂Ω) -ic, or bavararum  Koi{ imani oa o havasarumnerin, kareli  analiti{ koren arunakel min funkcia C 1 (Ω) -ic:

3. Hartogsi eorem. Aym berenq Hartogsin patkano s mek eorem harkadir analitik arunakuyan veraberyal:  e o r e m 2.3. Dicuq trva en G ⊂ Cmz , G0 ⊂ G tiruyne{

r (U ⊂ Cnw) bazmaglan Γ henqov: Nanakenq U ∗ = U ∪ Γ ( ) M = G × Γ ∪ G0 × U ∗ : Ee M -i vra orova f funkcian 1.

(

)

anndhat  G × Γ -i vra cankaca fiqsa ω -i hamar Γ -ic holomorf  G -um,

Lriv apacuyc tes [5]-um, j 53:

Glux ԼԼ. Holomorfuyan tiruyner

cankaca fiqsa z ∈ G0 keti hamar holomorf  U -um, apa ayn analitikoren arunakvum  G × U tiruyi mej: A p a c u y c: Ditarkenq 2.

F (z, w) = (2πi)n

∫ Γ

f (z, ω) dω ω−w

funkcian: Fiqsa z ∈ G -i hamar na holomorf  U -um fiqsa

w ∈ U -i hamar holomorf  G -um: st Hartogsi eoremi F - holomorf  G × U tiruyum: Bayc erb z ∈ G0 , apa norhiv 2 -rd paymani f - U -um nerkayacvum  ir Koii integralov, aynpes or ∫ f (z, w) =

(2πi)n

Γ

f (z, ω) dω = F (z, w) : ω−w

Ayspisov, F - analitikoren arunakum  tiruyi mej: §

f

funkcian

G×U

11. Holomorfuyan tiruyner

S a h m a n u m 2.2. Dicuq f1

funkcianer holomorf en hamapatasxanabar D1 D2 tiruynerum: Ee f1 ≡ f2 D1 ∩ D2 bazmuyan kapakcva komponenti vra, apa kasenq, or f2 - handisanum  f1 -i analitik arunakuyun D2 -i vra, isk f1 -` f2 -i arunakuyun D1 -i vra: Inpes haytni  mek kompleqs o oxakani funkcianeri te{ suyunic, amen mi D ⊂ C1 tiruyi hamar goyuyun uni funk{ cia, or holomorf  D -um analitikoren i arunakvum D -ic durs, ayl ba erov asa D -n nra hamar bnakan oroman ti{ ruy : Naxord paragrafum berva harkadir arunakman rinakner hangecnum en het yal sahmanman` f2

§ 11.

Holomorfuyan tiruyner

S a h m a n u m 2.3. D ⊂ Cn tiruy kovum  holomorfuyan

tiruy f funkciayi hamar, ee f - holomorf  D -um ana{

litikoren i arunakvum D -ic durs ver um nva imastov: D -n kovum  holomorfuyan tiruy, ee na holomorfuyan tiruy  or  funkciayi hamar: Sahmanumic hetuyamb bxum `  e o r e m 2.4. Ee D -n holomorfuyan tiruy  Cn -um,

isk G -n nman tiruy  Cm -um, apa D × G dekartyan ar{ tadryal holomorfuyan tiruy  Cn+m -um:

A p a c u y c: Vercnenq f funkcia, ori hamar D -n holo{

morfuyan tiruy  nman g funkcia G -i hamar: Ayd depqum f (z)g(z) ∈ O(D × G) D × G tiruy f (z)g(z) -i hamar klini holomorfuyan tiruy: 1. eorem argelqi veraberyal. Kasenq, or ζ ezrayin ke{ tum ka argelq, ee yuraqanyur M ⊂ D kompakt bazmuyan ε > 0 vi hamar goyuyun uni f ∈ O(D) aynpisin, or |f (z)| < 1 erb z ∈ M , bayc |f (z ′ )| > 1 in-or z ′ ∈ B(ζ, ε) ketum:  e o r e m 2.5 (argelqi veraberyal). Cankaca E ⊂ ∂D

bazmuyan hamar, ori yuraqanyur ketum ka argelq, go{ yuyun uni f ∈ O(D) , or ansahmana ak  E -i bolor ke{ terum: A p a c u y c: ntrenq E -um amenureq xit haveli E1 =

{ m }∞ ζ m=1

enabazmuyun: Bavakan  ka ucel f ∈ O(D) funkcia, or ansahmana ak  E1 -i keterum: E1 -i keter hamarakalenq aynpes, or amen mi ket handipi anverj qanakov angam: eorem klini apacucva , ee D -um gtnvi z m keteri hajordakanuyun holomorf f funkcia aynpisiq, or =

|z m − ζ m | → 0

f (z m ) → ∞ :

Glux ԼԼ. Holomorfuyan tiruyner

Aynuhet vercnenq Bm kompakt bazmuyunneri ndlaynvo n{ taniq, or spa um  D -n, aysinqn ∞ ∪

Bm ⊂ Bm+1

Bm = D :

m=1

Induktiv eanakov ka ucenq Bm -i Km enahajordakanu{ yun, z m keter D -ic fm ∈ O(D) funkcianer aynpisiq, or (a) |z m − ζ m | <

, m

(b) |fm (z m )| < 1 ,

erb

z ∈ Km ,

(c) |fm (z m )| > 1 :

Dra hamar nax vercnenq K1 = B1 : st argelqi sahmanman ζ 1 ketum, goyuyun unen f1 ∈ O(D) u z 1 ∈ D aynpisiq, or |z 1 − ζ 1 | < 1; |f1 (z)| < 1 erb z ∈ K1 , |f1 (z 1 )| > 1 : Aym enadrenq e ka ucum katarva  bolor k 6 m−1 veri hamar: Vercnenq m -n aynqan me , or ( }) ∪{ 1 Km ⊃ Km−1 z , . . . , z m−1 :

st argelqi sahmanman ζ m ketum, kgtnvi z m ∈ D ket fm ∈ ∈ O(D) funkcia aynpisiq, or bavararven (a) { (c) paymanner: Havi a nelov, or |fm (z m )| > 1 , kareli  ntrel bnakan pm veri aynpisi hajordakanuyun, or m−1 ∑ 1 m pm |f (z )| > |fk (z m )|pk + m : m m2 k2 k=1

Ays pndum hetuyamb apacucvum  indukciayov, nd orum, a ajin qayl aknhayt ` kvercnenq p1 = 1 : Aynuhet ditarkenq het yal arq ∞ ∑ f (z) = |f (z)|pk : (2.2) 2 k k=0

k

§ 11.

Holomorfuyan tiruyner

Ee z ∈ Km k > m , apa |fk (z)| < 1 , het abar, (2.2) arq Km -i vra havasaraa zugamet : Qani or Km -eri miacum spa um  amboj D -n, apa (2.2) arq havasaraa zuga{ mitum  D -i nersum st Vayertrasi eoremi nra gumar holomorf  D -um: Ev verjapes m−1 ∑ 1 m pm |f (z )| − |fk (z m )|pk − m m2 k2

|fm (z m )| >

∞ ∑

−

k=m+1

orteic er um , or

>m− k2

k=1 ∞ ∑

k=m+1

, k2

f (z m ) → ∞ :

Ayd eoremic het um en mi qani parz paymanner, oronc dep{ qum D -n holomorfuyan tiruy :

H e t a n q 2.1. Ee D tiruyi ezrayin bolor keterum

ka argelq, apa D -n holomorfuyan tiruy :

A p a c u y c: st argelqi masin eoremi goyuyun uni f ∈

∈ O(D) funkcia, or ansahmana ak  ezrayin bolor keterum: Het abar na i karo analitikoren arunakvel D -ic durs, uremn, D -n holomorfuyan tiruy :

H e t a n q 2.2. Ee D tiruyi ezrayin amen mi z0 keti

hamar goyuyun uni f ∈ O(D) , or ansahmana ak  z 0 -um (argelqi funkcia), apa D -n holomorfuyan tiruy :

A p a c u y c: Hamaayn 2.17 varuyan ezri bolor keterum

goyuyun uni argelq tiruy :

st het anq 2.1-i

D -n

H e t a n q 2.3. Yuraqanyur D ⊂ C1

fuyan tiruy :

holomorfuyan

tiruy holomor{

Glux ԼԼ. Holomorfuyan tiruyner A p a c u y c: Ee z 0 -n ezrayin ket , apa f (z) =

karo  a ayel orpes argelqi funkcia kira el het anq 2.2-:

z0

ketum,

-n z − z0

mnum 

H e t a n q 2.4. Amen mi g oren u ucik tiruy Cn -um

holomorfuyan tiruy :

A p a c u y c: G oren u ucik tiruyi sahmanumic hete{

vum , or goyuyun uni aynpisi l(z) = a1 z1 +· · ·+an zn +b g ayin funkcia, or {z : l(z) = 0} kompleqs hiperharuyun, ancnelov z 0 ketov, i hatvum D -i het: Owremn, orpes argelqi funkcia z 0 keti hamar kareli  vercnel f (z) = : l(z)

H e t a n q 2.5. Bazmaglan holomorfuyan tiruy : A p a c u y c: Ayd pndum het anq  ayn asti, or bazma{ glan g oren u ucik tiruy : Baci dranic, da het um  na eorem 2.4-ic:

§

12. Holomorf u ucikuyun

tara uyan mej D tiruyi sovorakan u ucikuyun kareli  sahmanel het yal  ov. ee K b D , apa K -i u ucik aan s kompaktoren patkanum  D -in: Isk K -i u ucik aan bakaca  ayn keteric, oroncum yuraqanyur g ayin funkciayi areqner en gerazancum nra maqsimumic K -i vra: Aydpisi motecum uyl  talis ndhanracnel u ucikuyan gaa ar funkcianeri tarber daseri hamar: Rn

§ 12.

Holomorf u ucikuyun

S a h m a n u m 2.4. Dicuq D -n tiruy  Cn -um, F - D -um

holomorf funkcianeri ntaniq 

K ⊂ D : Het

yal bazmuyun

{ } b = z ∈ D : |f (z)| 6 sup |f (ζ)|, ∀f ∈ F K ζ∈K

kovum 

K -i F -u ucik

S a h m a n u m 2.5.

aan:

tiruy kovum  F -u ucik (kam u ucik F ntaniqi nkatmamb), ee ayn banic, or K b D , het um , or Kb b D : D

Ee F = O(D) , apa F -u ucik tiruy kovum  holomorf u ucik: Isk ee F - hamnknum  g ayin funkcianeri, kam

bazmandamneri, kam l acional funkcianeri dasi het, apa F -u ucik tiruy kanvanenq hamapatasxanabar` g ayin, bazmandamayin, kam acional u ucik tiruy: Parz , or inqan aveli layn  F das, aynqan aveli ne  F -u ucik aan, het abar, aynqan aveli layn  F -u ucik tiruyneri ntaniq: Inpes het um  stor berva Kartani Tuleni erku eo{ remneric, tiruyi holomorf u ucikuyun anhraet  u ba{ varar, orpeszi na lini holomorfuyan tiruy:

 e o r e m 2.6 (Kartan{Tulen). Ee D ⊂ Cn

tiruy ho{ lomorf u ucik , apa na holomorfuyan tiruy : A p a c u y c: Apacucvum  argelqi masin eoremi nman:

Miayn e zm keter fm funkcianern ntrum enq, elnelov uri nkata umneric. D -i holomorf u ucikuyunic het um , or b m b D , uremn, goyuyun unen z m ∈ D keter gm ∈ O(D) K funkcianer aynpisiq, or |z m − ζ m | <

, m

|gm (z m )| > max |gm | : Km

Glux ԼԼ. Holomorfuyan tiruyner Orpes

fm

vercnum enq fm (z) =

gm (z) , max |gm |

m = 1, 2, . . . :

Km

Apacuyci mnaca mas mnum  nuyn` f (z) =

∞ ∑ |fm (z)|pm m2

m=0

funkcian holomorf  D -um D -i bolor ezrayin keterin mote{ nalis anverj aum , inic het um , or D -n holomorfuyan tiruy : D i t o  u  y u n 2.3. Parz , or eoremi paymani mej

holomorf u ucikuyun kareli r oxarinel u ucikuyamb st kamayakan F ⊂ O(D) ntaniqi: Kasenq, or D tiruyum holomorf funkcianeri F bazmu{ yun kazmum  das, ee amen mi f funkciayi het mekte na parunakum  na f -i bolor kargi a ancyalner af k -n, orte k -n kamayakan bnakan, isk a -n kamayakan kompleqs ver en: L e m m a 2.1 (miaamanakya arunakman veraberyal). Di{ cuq F - funkcianeri das , K b D ρ = ρ (K, ∂D) -n na{ nakum  polidiskayin metrikayov K -i he avoruyun ∂D ezric: Inpisin l lini Kb F aanin patkano a ket, kamayakan f ∈ F funkcia analitikoren arunakvum  U (a, ρ) polidiski mej: Baci dranic, bolor r < ρ hamar na bavararum  maqsimumi skzbunqin` sup |f (z)| 6 sup |f (z)|, z∈Kr

z∈U (a,r)

orte Kr =

∪ z∈K

U (z, r) :

(2.3)

§ 12.

gal

Holomorf u ucikuyun

Nenq, or ayste kar or , or U (a, ρ) polidisk karo  durs D tiruyi sahmanneric: A p a c u y c: Verlu enq f (z) - a keti rjakayqum eylori

arqi`

f (z) =

∞ ∑

(2.4)

ck (z − a)k :

|k|=0

Apacuyci hamar bavakan  cuyc tal, or ays arq U (a, ρ) -um zugamet , ayd depqum nra gumar kta pahanjveliq analitik arunakuyun: Dicuq r < ρ ; qani or Kr b D , apa nanakelov sup |f (z)| = Mf,r , Kr

st Koii anhavasaruyunneri kunenanq Mf,r 1 ∂ |k| f (z) 6 |k| , k k! ∂z r

z∈K:

Havi a nelov, or a ∈ Kb F or (2.4) arqi gor akicneri hamar 1 ∂ |k| f (a) ck = , stanum enq k k!

∂z

|ck | =

Aym vercnelov

Mf,r 1 ∂ |k| f (a) 1 ∂ |k| f (z) sup 6 |k| : k k k! ∂z ∂z r z∈K k! r1 < r ,

kamayakan

z ∈ U (a, r1 )

|ck (z − a)k | 6 Mf,r

( r )|k|

r

(2.5)

ketum

,

oric het um , or (2.4) arq U (a, r1 ) polidiskum maorvum  zugamet vayin arqov: Qani or r -n u r1 - kareli  vercnel

Glux ԼԼ. Holomorfuyan tiruyner

kamayakan a ov mot ρ -in, stacvum , or arq zugamet  amboj U (a, ρ) -um: (2.3)- apacucelu hamar bavakan  cuyc tal, or |f (z)| 6 Mf,r ,

(2.6)

z ∈ U (a, r1 )

kamayakan r1 < r depqum: Enadrenq (2.6)- it  in-or r1 < r hamar, aysinqn |f (z)| =α>1: z∈U (a,r1 ) Mf,r sup

Aysteic het um , or kamayakan bnakan [

f (z) φq (z) = Mf,r

q

vi hamar

]q

funkcian bavararum  sup

|φq (z)| = αq

z∈U (a,r1 )

havasaruyan: Myus komic, φq (z) apacucva (2.5) anhavasaruyan`

∈ F , Mφq ,r = 1

(2.7) , st

1 ∂ |k| φq (a) 6 |k| : k k! ∂z r

Owremn U (a, r1 ) polidiskum φq funkciayi hamar tei  unenum |φq (z)| 6

∞ ∑ 1 ∂ |k| φq (a) · |z − a||k| 6 k! ∂z k

|k|=0 ∞ ∑

|k|=0

1 ( r1 )|k| ( r1 )−n = 1− k! r r

§ 12.

Holomorf u ucikuyun

gnahatakan, or (2.7)-i het mekte hangecnum  ( r1 )−n αq 6 1 − r

anhavasaruyan kamayakan amboj q > 0 -i hamar: Bayc ayd anhavasaruyun hnaravor  bavakanaa me q -i depqum, orovhet α > 1 : Stacva hakasuyun apacucum  (2.3) anhavasaruyun kamayakan r1 < r -i depqum:

 e o r e m 2.7 (Kartan{Tulen). Kamayakan D ⊂ Cn holo{

morfuyan tiruy holomorf u ucik : A p a c u y c: Dicuq K b D

ρ = ρ (K, ∂D) : Qani or koordinatakan zk funkcianer,

O(D) -n parunakum  bolor apa Kb -n sahmana ak : Naxord lemmayic het um , or can{ kaca a ∈ Kb -i hamar U (a, ρ) ⊂ D , orovhet ka funkcia f ∈ O(D) , or i arunakvum analitikoren D -ic durs: Owremn b b D: K

eorem 2.6 eorem 2.7-ic het um 

 e o r e m 2.8. Orpeszi D ⊂ Cn tiruy lini holomorfu{

yan tiruy, anhraet  u bavarar, or na lini holomorf u ucik:

 e o r e m 2.9.

Orpeszi D ⊂ Cn tiruy lini holomorf u ucik, anhraet  u bavarar, or kamayakan K b D bazmuyan hamar ( ) b ∂D = ρ (K, ∂D) : ρ K,

A p a c u y c: Bavararuyun het um  sahmanumneric:

Anhraetuyun apacucelu hamar nax nkatenq, or mit ( ) b b ⊇ K : Ee ayste liner xist ρ K, ∂D 6 ρ (K, ∂D) , qani or K

Glux ԼԼ. Holomorfuyan tiruyner

anhavasaruyun, apa goyuyun kunenar Kb -in patkano a ket, ori hamar ρ (a, ∂D) < ρ (K, ∂D) : Ayd depqum, hamaayn miaamanakya arunakman veraberyal lemmayi, cankaca f ∈ ∈ O(D) funkcia karunakver a kentronov ρ (K, ∂D) a avov polidiski mej, aysinqn, D -ic durs, D -n r karo linel holomorfuyan tiruy:

 e o r e m 2.10. Dicuq Dα -n holomorfuyan tiruyneri ∩

kamayakan ntaniq  G = Dα : Ayd depqum G◦ bac korizi amen mi D kapakcva komponent holomorfuyan tiruy : A p a c u y c: Dicuq K b D , petq  cuyc tal, or Kb b D :

Amen mi funkcia O(Dα ) -ic holomorf  ⊃ O(Dα ) : Het abar b O(D) ⊂ K b O(D ) b D K α

D -um,

cankaca

α-i

aysinqn

O(D) ⊃

hamar:

Aynuhet ` b O(D) , ∂D) = inf ρ(K b O(D) , ∂Dα ) > inf ρ(K b O(D ) , ∂Dα ) = ρ(K α α

α

= inf ρ(K, ∂Dα ) = ρ(K, ∂D) > 0 : α

Ayste na havi  a nva eorem 2.1-ic bxo ρ(Kb O(Dα ) , ∂Dα ) = = ρ(K, ∂Dα ) havasaruyun: Nenq, or nman pndum holomorfuyan tiruyneri miavorman hamar it  (tes xndir 2.18): A anc apacuyci berenq bava{ rar payman, ori depqum holomorfuyan tiruyneri miavorum s holomorfuyan tiruy :

 e o r e m 2.11 (Behenke{ teyn). Holomorfuyan tiruy{

neri ndlaynvo hajordakanuyan

D1 b D2 b . . . b Dk b . . .

§ 12.

Holomorf u ucikuyun

miavorum s holomorfuyan tiruy :

 e o r e m 2.12.

Holomorfuyan tiruy linelu hatku{ yun invariant  biholomorf artapatkerumneri nkat{ mamb: A p a c u y c: Petq  apacucel, or ee D ⊂ Cn holomorfu{

yan tiruy  D∗ - nra patkern  φ biholomorf artapat{ kerman amanak, apa D∗ -n s holomorfuyan tiruy : c∗ O(D∗ ) b D∗ : Nkatenq, or Dicuq K ∗ b D∗ : Cuyc tanq, or K ee K ∗ b D∗ K = φ−1 (K ∗ ) , apa φ -i homeomorfuyan norhiv K b D : Ayd depqum b O(D) b D, K (2.8) orovhet

D -n

holomorf u ucik : Aym hanozvenq, or ( ) c∗ O(D∗ ) : b O(D) ⊃ K φ K

Vercnenq

( ) b O(D) w0 ∈ D∗ \ φ K

(2.9)

ket: Ayd depqum

b O(D) z 0 = φ−1 (w0 ) ∈ D \ K

goyuyun uni f ∈ O(D) funkcia aynpisin, or |f (z 0 )| > > sup |f | : Ditarkenq ψ(w) = f ◦ φ−1 (w) : Parz , or ψ ∈ O(D∗ ) K

|ψ(w0 | > sup |ψ| :

c∗ O(D∗ ) : Owremn, w0 ̸∈ K Aynuhet , (2.8)-ic φ -i homeomorf lineluc het um , or b c∗ O(D∗ ) b D∗ st (2.9)-i: Ayspisov, D∗ -n φ(KO(D) ) b D∗ K holomorf u ucik  , het abar, holomorfuyan tiruy : K∗

Glux ԼԼ. Holomorfuyan tiruyner §

13. Ow ucikuyun st L ii

Kompleqs r -a ani analitik S maker uy Cn tara uyu{ num kovum  ∆ ⊂ Cr tiruyi patker φ = (φ1 , . . . , φn ) : ∆ 7→ Cn

(n > r) ) ( ∂φi Yakobii ∂ζj

matrici artapatkerman amanak, orte

ang amenureq ∆ -um havasar  r -i: Masnavorapes, kompleqs mek a ani analitik maker uy anvanum en na analitik kor: S - kovum  kompakt analitik maker uy, ee na mi uri r -a ani analitik maker uyi kompakt enabazmuyun : Kompakt analitik maker uyneri hamar tei uni maqsimumi skzbunq. ee f funkcian holomorf  S -um anndhat  nra akman vra, apa

max |f | = max |f | : S

∂S

 e o r e m 2.13 (Behenke{Zomer).

Dicuq Sk -er kompakt analitik maker uyner en, oronq patkanum en D tiruy{ in irenc ∂Sk ezreri het miasin: Ee Sk hajordakanuyu{ n zugamitum  in-or S bazmuyan, isk ∂Sk -n` Γ -in Γ b D , apa amen mi funkcia f ∈ O(D) analitikoren a{ runakvum  S -i in-or rjakayq: A p a c u y c: Qani or Γ b D , apa goyuyun uni G b D

aynpisi tiruy, or Γ b G : Nanakenq ρ(G, ∂D) = r : Qani or goyuyun uni k0 , or erb k > k0

∂Sk → Γ ,

(2.10)

∂Sk ⊂ G :

st maqsimumi skzbunqi kamayakan

f ∈ O(D)

|f (z)| 6 max |f |, ∂Sk

z ∈ Sk

hamar

§ 13.

Ow ucikuyun st L ii

, havi a nelov (2.10)-, stanum enq |f (z)| 6 max |f | : G

Da nanakum , or z ket, het abar amboj Sk -n, patka{ num  GbO(D) u ucik aanin, erb k > k0 : Miaamanakya arunakman veraberyal lemma 2.1-ic het um , or amen mi f ∈ ∈ O(D) funkcia analitikoren arunakvum  Sk maker uy{ neri Skr r -rjakayqi mej: Aynuhet , qani or Sk → S , goyuyun uni k1 > k0 aynpisin, or S ⊂ Skr/2 bolor k > k1 hamar , het abar, yuraqanyur r/2 f ∈ O(D) analitikoren arunakvum  Sk -i mej: Apacuycic er um , or O(D) -i oxaren kareli∪ r vercnel funkcianeri das, oronq holomorf en sahmanayin S Γ bazmu{ yan or  rjakayqi D -i hatman vra:

S a h m a n u m 2.6.

tiruy kovum  u ucik st L ii (kar` L -u ucik) ezrayin ζ ketum, ee, inpisin l lini ζ -n parunako S maker uy, aynpisin, or ∂S ⊂ D , kamayakan Sk analitik maker uyneri hajordakanuyun, ori hamar D ⊂ Cn

Sk → S,

∂Sk → ∂S,

goyuyun uni k0 hamar, oric sksa bolor Sk -er parunakum en D -in patkano keter:

 e o r e m 2.14. Ee D tiruyi or

 ζ ezrayin ket ka{ reli  drsic oa el analitik S = {z ∈ Cn : f (z) = 0} baz{ muyamb, orte f - ζ -um holomorf funkcia , apa D -n L -u ucik : A p a c u y c: eoremi paymanic het um , or goyuyun uni

f

funkcia, or holomorf  ζ keti or  Uζ rjakayqum, havasar

Glux ԼԼ. Holomorfuyan tiruyner

∩

 zroyi ayd ketum tarber  zroyic Uζ D -um: Ee D -n L -u u{ cik liner ζ -um, apa goyuyun kunenar aynpisi S maker uy, or ζ ∈ S , ∂S ⊂ D , Sk kompakt analitik maker uyneri hajordakanuyun, or Sk ⊂ D , Sk → S , ∂Sk → ∂S : Ayd depqum kamayakan f ∈ O(D) analitikoren karunakver ζ ket,∩isk da khakaser ayn banin, or g = 1/f funkcian holomorf  Uζ D -um i arunakvum ζ -i rjakayq:

Xndirner X n d i r 2.1. Dicuq f - holomorf 

E ⊂ Cn polidiskum anndhat  E ∪ Γ bazmuyan vra, orte Γ -n E -i henqn : Apacucel, or f - anndhatoren arunakvum  E -i vra:

X n d i r 2.2. Dicuq L - l(z) = c0 + c1z1 + · · · + cnzn g ayin

funkcianeri ntaniqn : Cuyc tal, or D tiruy L -u ucik  ayn miayn ayn depqum, erb na u ucik  sovorakan erkraa{

akan imastov: X n d i r 2.3. Dicuq M - czk = cz1k1 · · · znkn bolor mian{ damneri ntaniqn  ( ki -er o bacasakan amboj ver en, c -n` kompleqs hastatun ): Cuyc tal, or 0 kentronov eynharti lriv tiruy logarimoren u ucik  ayn miayn ayn depqum, erb na M -u ucik : X n d i r 2.4. Apacucel, or C haruyan patkano K kompakt bazmandamayin u ucik  ayn miayn ayn depqum, erb C \ K bazmuyun kapakcva : X n d i r 2.5. Apacucel, or kamayakan K ⊂ Cn kompakti

acional u ucik aan hamnknum  A = {z ∈ Cn : P (z) ∈ P (K)

bolor

P

bazmandamneri hamar}

Xndirner

bazmuyan het:

X n d i r 2.6. Dicuq K -n kompakt bazmuyun 

Cn -um:

X n d i r 2.7. Dicuq K -n kompakt bazmuyun 

Cn -um,

Apacucel, or P (K) hanrahavi bolor anndhat g ayin multi{ plikativ funkcionalneri M tara uyun kareli  nuynacnel b bazmandamayin u ucik aani het het yal imastov. can{ K kaca m funkcional M -ic irenic nerkayacnum  <areq z 0 ketum>, z 0 ∈ Kb , aysinqn, m(f ) = f (z 0 ) cankaca f -i hamar P (K) -ic: ori hamar u ucik :

P (K) = C(K) :

Apacucel, or

K -n

bazmandamayin

X n d i r 2.8. Apacucel, or Cn -i irakan haruyan patka{

no kamayakan kompakt bazmuyun bazmandamayin u ucik :

X n d i r 2.9. Cuyc tal, or bazmandamnerov orovo D = {z ∈ Cn : |Pm (z)| < 1, m = 1, · · · , N }

bazmanist bazmandamayin u ucik tiruy :

X n d i r 2.10. Dicuq δ ∈ (0, 2π)

M -

{

} z ∈ C2 : z1 = eit , δ 6 t 6 2π, z2 = 0 { } z1 = eit , 0 6 t 6 δ, |z2 | = 1

bazmuyunneri miavorumn : Apacucel, or {

z ∈ C2 : |z1 | 6 1, z2 = 0

rjan parunakvum  mej:

M -i

}

bazmandamayin u ucik aani

Glux ԼԼ. Holomorfuyan tiruyner

X n d i r 2.11. Dicuq K -n { } z ∈ C2 : |z|2 = 2

gndolorti

{

z ∈ C2 : z2 = z̄1

haruyan hatumn : Apacucel, or

X n d i r 2.12. Apacucel, or

}

P (K) = C(K) :

B = {z ∈ Cn : |z| < 1}

ilovi ezr hamnknum  nra topologiakan ezri het:

gndi

X n d i r 2.13. Apacucel, or Cn -um E polidiski ilovi S(E)

ezr Bergmani

B(E)

ezr hamnknum en nra henqi het:

X n d i r 2.14. Dicuq { } D = z ∈ C2 : 0 < |z1 | < 1, |z2 | < |z1 | :

Cuyc tal, or a)

D -n

holomorfuyan tiruy ,

b)

D -n

holomorfuyan tiruyneri hatum :

X n d i r 2.15. Apacucel, or

{ } D = z ∈ C2 : 0 < |z1 | < 1, |z2 | < |z1 |− ln |z1 |

tiruyi ilovi Bergmani ezrer iraric tarber en. { } S(D) = z ∈ C2 : |z1 | < 1, |z2 | = |z1 |− ln |z1 | ,

isk

{ } B(D) = z ∈ C2 : |z1 | = |z2 | = 1 :

Xndirner

X n d i r 2.16. Dicuq D1 - D2 - haruyan vra oork korerov sahmana akva tiruyner en, oronq asta en ko{ ordinatneri skzbnaketi nkatmamb, K = {tz : 0 6 t 6 1, z ∈ ∂D1 × ∂D2 } :

Apacucel, or K -i rjakayqum kamayakan holomorf f funkcia analitikoren arunakvum  D1 × D2 tiruyi vra:

X n d i r 2.17. Dicuq goyuyun uni f ∈ O(D) , or ansahma{

na ak  ζ -um, aysinqn, goyuyun uni aynpisi z m ∈ D keteri hajordakanuyun, or lim z m = ζ lim f (z m ) = ∞ : Cuyc tal, or ayd depqum ζ -um ka argelq:

X n d i r 2.18. Berel rinak, erb erku holomorfuyan ti-

ruyneri miavorum holomorfuyan tiruy :

X n d i r 2.19. Apacucel, or { } D = z ∈ C2 : |z1 |2 + x22 > ρ2

tiruy holomorfuyan tiruy :

X n d i r 2.20. Cuyc tal, or Kuzeni a ajin problem o mit

lu um uni

{ } { } D = z ∈ C2 : |z1 | < 3, |z2 | < 3 \ z ∈ C2 : |z1 | < 3, 1 < |z2 | < 3

krknaki rjana tiruyum, orn, uremn, holomorfuyan ti{ ruy :

GLUX

PSEVDOU UCIK TIRUYNER

Ays glux nvirva  plyurisubharmonik (masnavorapes, u ucik) funkcianerin nranc hamapatasxan ps dou ucik (masnavorapes, u ucik) tiruynerin: Hajord glxum menq khamozvenq, or holomorfuyan tiruyner usumnasirelis plyurisubharmonik funkcianer a ayum en orpes himnakan gor iq : Mi qani kompleqs o oxakani plyurisubharmonik funk{ cianer sahmanvum en mek kompleqs o oxakani (kam, or nuynn , erku irakan o oxakani) subharmonik funkciane{ ri mijocov: Ev, uremn, bnakan  nax usumnasirel subharmonik funkcianer:

§

14. Subharmonik funkcianer

1. Kisaanndhat funkcianer. Hetagayi hamar kar or

der en katarum ayspes kova kisaanndhat funkcianer, oronc sahmanumn u oro hatkuyunner berva en stor : Dicuq D ⊂ Cn tiruyum orova  irakan u(z) funkcia: Ee z 0 ∈ D ketum u(z 0 ) = lim u(z) = lim sup u(z), z→z 0

apa ee

u(z) -

δ→0 B(z 0 ,δ)

kovum  kisaanndhat ver ic

z 0 -um:

u(z 0 ) = lim u(z) = lim inf u(z), z→z 0

δ→0 B(z 0 ,δ)

Nmanapes,

§ 14.

Subharmonik funkcianer

apa u(z) - kisaanndhat  nerq ic: Masnavorapes, ee u(z 0 ) = +∞ ( u(z 0 ) = −∞ ), apa u(z) - kisaanndhat  ver ic (nerq ic) z 0 ketum: Nenq, or ee u(z) - z 0 -um kisaanndhat  ver ic, ner{ q ic |u(z 0 )| ̸= ∞ , apa ayn anndhat  ayd ketum: Ee u(z) - kisaanndhat  nerq ic, apa −u(z) - kisa{ anndhat  ver ic , uremn, bavakan  usumnasirel, rinak, ver ic kisaanndhat funkcianer: Kasenq, or u(z) - kisaanndhat  ver ic D -um, ee ayn kisaanndhat  ver ic D -i bolor keterum: varkenq ver ic kisaanndhat funkcianeri mi qani hat{ kuyun, oronc apacuyc onum enq nercoin: •

Ee |u(z 0 )| ̸= ∞ , apa u(z) - kisaanndhat  ver ic ayn miayn ayn depqum, erb kamayakan ε > 0 vi hamar goyuyun uni aynpisi δ > 0 , or |z − z 0 | < δ ⇒ u(z) < u(z 0 ) + ε :

•

Orpeszi u funkcian lini ver ic kisaanndhat, anhra{ et  u bavarar, or kamayakan a ∈ (−∞, +∞) vi hamar {z ∈ D : u(z) < a} bazmuyun lini bac:

•

Ee u funkcian ver ic kisaanndhat  K kompakti vra u(z) ̸= +∞ , apa na K -i vra ver ic sahmana ak  ndunum  ir me aguyn areq:

2. Subharmonik funkciayi sahmanum. G ⊂ C1 tiruy{

um orova

u(z)

funkcian kovum  subharmonik, ee

1.

−∞ 6 u(z) < +∞ ,

2.

u(z) -

kisaanndhat  ver ic G -um,

Glux ԼԼԼ. Ps dou ucik tiruyner

3. kamayakan G′ ⊂ G enatiruyi kamayakan h(z) funk{ ciayi hamar, or harmonik  G′ -um anndhat  G′ -um, u(z) 6 h(z) ∂G′ -i vra paymanic het um , or u(z) 6 h(z) , erb z ∈ G′ : Ee −u(z) funkcian subharmonik , apa u(z) - kovum  su{ perharmonik: Harmonik funkcian miaamanak subharmonik, superharmonik funkcia : D i t o  u  y u n 3.4. Menq ditarkum enq erku irakan

o oxakani subharmonik funkcianer: Hamapatasxan sah{ manumn u hatkuyunner a anc akan o oxuyunneri mnum en ui mej na cankaca vov o oxakani subharmonik funk{ cianeri hamar: 3. Harnaki eorem. D tiruyum hk (z) nvazo, harmo{ nik funkcianeri hajordakanuyan sahman kam harmo{ nik funkcia  D -um, kam nuynabar −∞ : A p a c u y c: Dicuq U (z 0 , r) b G : Inpes haytni , U (z 0 , r)

rjanum harmonik nra akman vra anndhat amen mi h(z) funkcia nerkayacvum  Puasoni integralayin bana ov` h(z) = 2π

∫2π P (z − z 0 , reiθ )h(z 0 + reiθ )dθ,

z ∈ U (z 0 , r),

(3.1)

orte P (z, ζ) =

r 2 − ρ2 , r2 − 2rρ cos(φ − θ) + ρ2

ζ = reiθ , z = ρeiφ

Puasoni korizn : Aysteic hetuyamb stacvum  het yal anhavasaruyun` r−ρ r+ρ 6 P (z, ζ) 6 , r+ρ r−ρ

(3.2)

§ 14.

Subharmonik funkcianer

mijin areqi eorem` h(z) = 2π

∫2π

(3.3)

h(z + reiθ )dθ :

Kira elov (3.1) Puasoni bana  hk −hk+m funkcianeri nkat{ mamb gtvelov (3.2) u (3.3) hatkuyunneric, stanum enq Har{ naki anhavasaruyun` ] r−ρ[ hk (z 0 ) − hk+m (z 0 ) 6 hk (z) − hk+m (z) 6 r+ρ ] r+ρ[ hk (z 0 ) − hk+m (z 0 ) , r−ρ

(3.4)

or it  bolor z ∈ U (z 0 , r) keteri hamar: Dicuq hk (z 0 ) → −∞ : (3.4)-ic het um , or hk (z) → −∞ havasaraa U (z 0 , r/2) -um: Aysteic Hayne{Boreli lemmayic ezrakacnum enq, or hk (z) → −∞ havasaraa amen mi G′ b G enatiruyum, aysinqn, havasaraa G -um: Isk aym enadrenq hk (z 0 ) → a > −∞ : st apacuca i, h(z) = lim hk (z) > −∞ G -um: (3.4)-ic het um , or hk (z) → h(z) havasaraa U (z 0 , r/2) -um: st Hayne{Boreli lemmayi, hk (z) - zugamitum  h(z) -in havasaraa amen mi G′ b G enati{ ruyum: Owremn, h(z) - harmonik  G -um: 4. Subharmonikuyan haytani: Dicuq u(z) funkcian

subharmonik  G tiruyum u(z) 6 2π

U (z 0 , r) b G :

Ayd depqum

∫2π P (z − z 0 , reiθ )u(z 0 + reiθ )dθ,

z ∈ U (z 0 , r) :

(3.5)

Haka ak, ee G tiruyum ver ic kisaanndhat u(z)

Glux ԼԼԼ. Ps dou ucik tiruyner

funkcian bavararum  u(z) 6 2π

∫2π u(z + reiθ )dθ,

z∈G

(3.6)

anhavasaruyan bolor bavakanaa

oqr r < r0 ha{ mar, apa u -n subharmonik  G -um: A p a c u y c: Dicuq u -n subharmonik  G -um: Qani or na ver ic kisaanndhat  U (z 0 , r) -um, apa goyuyun uni annd{ hat uk funkcianeri nvazo hajordakanuyun, or U (z 0 , r) -i vra zugamitum  u -in (tes xndir 3.1): Nanakenq hk -ov uk -i harmonik arunakuyun U (z 0 , r) rjani vra: Qani or hk+1 (ζ) = uk+1 (ζ) 6 uk (ζ) = hk (ζ),

z ∈ ∂U (z 0 , r),

apa, st maqsimumi skzbunqi` hk+1 (z) 6 hk (z),

z ∈ U (z 0 , r) :

(3.7)

Aysteic st Harnaki eoremi ezrakacnum enq, or h∗ (z) = lim hk (z) k→∞

(3.8)

funkcian kam harmonik  U (z 0 , r) -um, kam nuynabar havasar  −∞ : Qani or u -n subharmonik  G -um u(z) 6 uk (z) = hk (z),

apa

u(z) 6 hk (z),

z ∈ ∂U (z 0 , r),

(3.9)

z ∈ U (z 0 , r) :

Aysteic (3.8)-ic` u(z) 6 h∗ (z),

z ∈ U (z 0 , r) :

(3.10)

§ 14.

Subharmonik funkcianer

Aym havi a nelov (3.1), (3.7){(3.9) L ii eorem, stanum enq h (z) = lim hk (z) = lim k→∞ k→∞ 2π ∗

∫2π P (z − z 0 , reiθ )hk (z 0 + reiθ )dθ =

=

=

=

2π 2π 2π

∫2π P (z − z 0 , reiθ ) lim hk (z 0 + reiθ )dθ = k→∞

∫2π

P (z − z 0 , reiθ ) lim uk (z 0 + reiθ )dθ = k→∞

∫2π

P (z − z 0 , reiθ )u(z 0 + reiθ )dθ :

(3.11)

Aysteic (3.10)-ic bxum  pahanjveliq (3.5)-: Haka ak, dicuq u(z) < +∞ funkcian ver ic kisaannd{ hat  G -um bavararum  (3.6) anhavasaruyan: Dicuq h - harmonik  G′ b G -um, anndhat  G′ -um h(z) > u(z) ∂G′ -i vra: Enadrenq goyuyun uni z ′ ∈ G′ ket, or h(z ′ ) < u(z ′ ) : Ayd depqum f (z) = u(z) − h(z) funkcian ver ic kisaanndhat  ′ G -um, o drakan  ∂G′ -i vra drakan  z ′ ∈ G′ ketum: Owremn, na hasnum  ir M > 0 me aguyn areqin or  z 0 ∈ G′ ketum: norhiv (3.3)-i f (z) - bavararum  (3.6) anhavasaruyan bo{ lor bavakanaa

oqr r 6 r0 (z 0 ) hamar: Ayd depqum f (z) ≡ ≡ M U (z 0 , r) -um: Iroq, ee or  z ′ ∈ U (z 0 , r) ketum f (z) < M , apa ver ic kisaanndhatuyunic khet er, or ayd anhavasa{ ruyun pahpanvum  z ′ -i or  rjakayqum, isk da khakaser (3.6)-in: st Hayne{Boreli lemmayi, kareli  nel aynpisi r0 = r0 (G′ ) iv, or (3.6)- tei unena bolor z ∈ G′ r 6 r0 hamar: Kira{

elov ayd anhavasaruyun G′ tiruyi ayn z keteri hamar, oronc hamar f (z) = M , kstananq f (z) ≡ M > 0 G′ -um, in

Glux ԼԼԼ. Ps dou ucik tiruyner

hakasum  f (z) 6 0 paymanin ∂G′ -i vra: Stacva hakasu{ yun apacucum , or h(z) > u(z) G′ -um: Ayspisov, u(z) - subharmonik  G -um: 5. Amena oqr harmonik maorant. Ee u(z) funkcian

subharmonik  U (z 0 , r) -um ver ic kisaanndhat  U (z , r) -um, apa § 11.4 eoremi apacucman nacqum ka uca h∗ (z) funkcian harmonik  U (z 0 , r) -um ver ic kisaanndhat  U (z 0 , r) -um: norhiv (3.11)-i, na kaxva

 uk → u hajordakanuyan ntruyunic: h∗ (z) funkcian kovum  u(z) funkciayi amena oqr harmonik maorant U (z 0 , r) rjanum: Na uni het yal hatkuyun. ee h funkcian harmonik  U (z 0 , r) -um, anndhat  U (z 0 , r) -um u(z) 6 h(z) ∂U (z 0 , r) -i vra, apa h∗ (z) 6 h(z) U (z 0 , r) -um: Iskapes, havi a nelov (3.1)- (3.11)-, stanum enq h (z) = 2π ∗

∫2π P (z − z 0 , reiθ )u(z 0 + reiθ )dθ 6

2π

∫2π P (z − z 0 , reiθ )h(z 0 + reiθ )dθ = h(z) :

Nman  ov sahmanvum  G tiruyum subharmonik u(z) funk{ ciayi amena oqr harmonik maorant, ee ∂G ezr aynpisin , or ∂G -i vra trva kamayakan anndhat funkciayi hamar Di{ rixlei xndir uni lu um: 6. Subharmonik funkcianeri parzaguyn hatkuyunner.

Subharmonik funkciayi sahmanumic § 11.4 -um trva hayta{ niic bxum en het yal hatkuyunner.

1.

2.

3.

4.

§ 14.

Subharmonik funkcianer

Ee funkcian subharmonik  G tiruyi yuraqanyur ke{ ti rjakayqum, apa ayn subharmonik  G -um: Subharmonik funkcianeri drakan gor akicnerov g ayin kombinacian s subharmonik : Subharmonik funkcianeri hajordakanuyan havasara{ a sahman subharmonik : Subharmonik funkcianeri monoton nvazo uk hajor{ dakanuyan sahman subharmonik :

Iroq, st kisaanndhat funkcianeri hatkuyunneri, kamayakan G′ b G enatiruyi hamar goyuyun uni aynpisi hastatun C < +∞ , or uk+1 (z) 6 uk (z) 6 C : Owremn, lim uk (z) funkcian kisaanndhat  ver ic: Anc{ k→∞ nelov (3.6)-um sahmani, st L ii eoremi kstananq lim uk (z) 6 k→∞ 2π

∫2π lim uk (z + reiθ )dθ :

k→∞

st § 11.4 -i haytanii nik : 5.

lim uk (z)

k→∞

funkcian subharmo{

Ee subharmonik funkcian ndunum  ir me aguyn areq tiruyi nersum, apa ayn nuynabar hasta{ tun :

Iroq, ee u(z) funkcian nduner ir me aguyn areqin z 0 ∈ G ketum, apa, inpes § 11.4 -i hakadar eoremi apacucman amanak, kstanayinq u(z) ≡ M G -um: 6.

Ee uα subharmonik funkcianeri ntaniqi u(z) = sup uα (z) α

Glux ԼԼԼ. Ps dou ucik tiruyner

verin paruri ver ic kisaanndhat  G -um, apa ayn subharmonik : A p a c u y c: Dicuq r < ∆G (z) : Kira elov (3.6) anhava{

saruyun yuraqanyur stananq u(z) = sup uα (z) 6 2π α

∫2π

uk (z)

funkciayi nkatmamb, k{

sup uα (z + re ) = 2π α

∫2π

iθ

u(z + reiθ )dθ :

st § 11.4 -i haytanii u(z) funkcian subharmonik : 7. Subharmonik funkciayi mijin areq: Dicuq u(z) -

subharmonik 

G

tiruyum: Het yal funkcian`

J(r, z ; u) = 2π

∫2π

u(z 0 + reiθ )dθ

kovum  u(z) -i mijin areq: Parz , or na orova  bolor ayn { } r -eri hamar, oronc hamar z : |z − z 0 | = r ⊂ G : { }

Ee u(z) - subharmonik  G -um z : |z − z 0 | < R ⊂ G , apa nra mijin areq aum  st r -i [0, R) -um: A p a c u y c: Dicuq r1 < r2 kamayakan ver en [0, R) -ic

- u(z) -i amena oqr harmonik maorantn  rjanum: Ayd depqum

h∗ (z)

|z − z 0 | < r2

J(r1 , z 0 ; u) 6 J(r1 , z 0 ; h∗ ) = J(r2 , z 0 ; h∗ ) = J(r2 , z 0 ; u) :

8. Subharmonik funkciayi rinakner. Ee u(z) funkci{ an subharmonik  G tiruyum, apa eu(z) - s subharmonik , isk ee lracuci u(z) > 0 , apa up (z) ( p > 1 ) funkcian nuynpes subharmonik  (tes xndirner 3.2 3.3):

§ 15.

Plyurisubharmonik funkcianer

Dicuq f (z) funkcian holomorf  G -um: Ayd depqum het yal funkcianer` |f (z)|p = ep ln |f (z)| ,

ln+ |f (z)| = max (0, ln |f (z)|)

subharmonik en: norhiv ver um nva i § 11.6 -i, ays pndum bavakan  apacucel ln |f (z)| -i hamar: Parz , or ayd funkcian ver ic kisaanndhat : Ayn keterum, orte ln |f (z)| = −∞ , (3.6) anhavasaruyun aknhaytoren bavararvum , mnaca

keteri rjakayqum ln |f (z)| = Re ln f (z) funkcian harmonik  (3.6)- bavararvum  st (3.3)-i: Ayspisov, ee f - holomorf  U (0, r) -um, apa (3.5)-ic bxum  Yenseni anhavasaruyun` ln |f (z)| 6 2π

∫2π P (z, reiθ ) ln f (reiθ ) dθ :

§

15. Plyurisubharmonik funkcianer

Hetagayi hamar kar or der  katarum ayspes kova he a{ voruyan funkcian, ori sahmanum hatkuyunner bervum en stor : 1. He avoruyan funkcian: Dicuq O -n bac bazmuyun 

-um: Nanakenq ∆O (x) -ov he avoruyan funkcian, ay{ sinqn, x ∈ O keti he avoruyun ∂O ezric`

Rn

∆O (x) = sup r,

orte

B(x, r) ⊂ O :

Ee O ̸= Rn , apa ∆O (x) - anndhat funkcia  O -um:

Glux ԼԼԼ. Ps dou ucik tiruyner

Iroq, ∆O (x) funkcian verjavor  O -um: Dicuq |x′ −x′′ | < ε : Ayd depqum |x − x′ | < |x − x′′ | + ε : Ee |x − x′ | < ∆O (x′ ) − ε , kstananq |x − x′ | < ∆O (x′ ) : Aysteic het um , or x ∈ O , st sahmanman, ∆O (x′′ ) > ∆O (x′ ) − ε : oxelov x′ x′′ keteri derer, stanum enq ∆O (x′ ) > ∆O (x′′ )−ε : Owremn, ee |x′ −x′′ | < ε , |∆O (x′ ) − ∆O (x′′ )| 6 ε : Ee A ⊂ O , apa ∆O (A) -ov nanakvum  A -i he avoruyun ∂O -ic` ∆O (A) = inf ∆O (x) : x∈A

Aknhayt , or ∆O

(∪

) Aα = inf α ∆O (Aα ) :

α

Aysteic u Hayne{Boreli lemmayic het um , or ee A b O , apa ∆O (A) > 0 , isk ee A -n sahmana ak  ∆O (A) > 0 , apa A b O: Dicuq Oα -n ∪(α = 1, 2, . . .) bac bazmuyunneri hajorda{ kanuyun  Oα = O : Kamayakan A b O bazmuyan α hamar goyuyun uni aynpisi N = N (A) iv, or ∆Oα (A) 6 ∆Oα+1 (A) 6 ∆O (A),

erb α > N,

(3.12)

lim ∆Oα (A) = ∆O (A) :

α→∞

Apacuyc anmijapes bxum  sahmanumneric: Dicuq O -n Oα bac bazmuyunneri hatman nersn : Ka{ mayakan A ⊂ O bazmuyan hamar

(3.13) Ays havasaruyun apacucelu hamar bavakan  ayn apa{ ( ∩ )◦ cucel A = {x} bazmuyan Oα bazmuyan amen mi B α kapakcva komponenti hamar, ayn ` ∆O (A) = inf ∆Oα (A) : α

∆B (x) = inf ∆Oα (x) : α

§ 15.

Qani or

Plyurisubharmonik funkcianer

x ∈ B ⊂ Oα ,

apa

∆B (x) 6 ∆Oα (x)

, uremn

∆B (x) 6 inf ∆Oα (x) : α

Myus komic, qani or

( ) B x, ∆Oα (x) ⊂ Oα

x ∈ B,

apa

( ) B x, inf ∆Oα (x) ⊂ B : α

Aysteic het um  inf ∆Oα (x) 6 ∆B (x) α

anhavasaruyun, or naxordi hakadarn , inic pndum:

bxum 

Dicuq G -n tiruy  Cn -um, a -n kompleqs vektor , |a| = 1 , λ -n kompleqs parametr : Nanakenq ∆a,G (z) -ov z keti he{

avoruyun min G tiruyi z ′ = z + λa erka anali{ tik haruyan hatum, orn irenic nerkayacnum  erka bac bazmuyun, aynpes or ∆a,G (z) = sup r,

ee

{

} z ′ : z ′ = z + λa, |λ| < r ⊂ G :

∆a,G (z) funkcian G -um kisaanndhat  nerq ic kamaya{ kan a -i depqum:

Nermu elov Gz,a = {λ : z + λa ∈ G} bazmuyun

o oxakani haruyan vra, karo enq grel ∆a,G (z) = ∆Gz,a (0) :

Aknhayt , or ∆G (z) = inf ∆a,G (z) : |a|=1

λ

kompleqs

Glux ԼԼԼ. Ps dou ucik tiruyner

2. Plyurisubharmonik funkciayi sahmanum: u(z) fun{

kcian kovum  plyurisubharmonik 1.

u(z) -

kisaanndhat  ver ic

D ⊂ Cn

tiruyum, ee

D -um,

2. kamayakan z 0 ∈ D keti hamar nra hetq ayd ketov anc{ no kamayakan kompleqs ui vra subharmonik : Aveli manramasn erkrord payman nanakum , or kamaya{ kan a ∈ Cn vektori hamar u(z 0 + λa) funkcian subharmonik  st λ -i { } Dz 0 ,a = λ ∈ C : z 0 + λa ∈ D

bac bazmuyan amen mi kapakcva komponenti vra: Ee −u(z) funkcian plyurisubharmonik , apa u(z) - ko{ vum  plyurisuperharmonik: Plyuriharmonik funkcian mia{ amanak plyurisubharmonik , plyurisuperharmonik: Orpes plyurisubharmonik funkcianeri rinak karo en a{

ayel ln |f (z)| , ln+ |f (z)| |f (z)|p funkcianer, orte f (z) - holomorf  G -um (tes § 11.8 -): 3. Plyurisubharmonikuyan haytani: Orpeszi D ti{ ruyum ver ic kisaanndhat u(z) funkcian lini plyuri{ subharmonik, anhraet  bavarar, or yuraqanyur z ∈ D keti ω ∈ Cn vektori hamar goyuyun unena aynpisi r0 = r0 (z, ω) iv, or u(z) 6 2π

∫2π

( ) u z + ωreit dt

erb r < r0 :

(3.14)

Ayd pndum anmijapes het um  haytaniic:

§ 12.2

sahmanumic

§ 11.4

§ 15.

Plyurisubharmonik funkcianer

4. Plyurisubharmonik funkcianeri parzaguyn hatku{ yunner. Stor varka hatkuyunner het um en sub{

harmonik funkcianeri hamapatasxan hatkuyunneric: 1.

2.

3.

4.

5.

6.

Ee funkcian plyurisubharmonik  D tiruyi yuraqan{ yur keti rjakayqum, apa ayn plyurisubharmonik  D -um: Plyurisubharmonik funkcianeri drakan gor akicnerov g ayin kombinacian s plyurisubharmonik : Plyurisubharmonik funkcianeri hajordakanuyan ha{ vasaraa sahman plyurisubharmonik : Plyurisubharmonik funkcianeri monoton nvazo hajor{ dakanuyan sahman plyurisubharmonik : Ee plyurisubharmonik funkcian ndunum  ir me aguyn areq tiruyi nersum, apa ayn nuynabar hastatun : Ee φα plyurisubharmonik funkcianeri ntaniqi verin paruri` u(z) = sup uα (z) α

ver ic kisaanndhat  nik :

D -um,

apa ayn plyurisubharmo{

5. Plyurisubharmonik funkciayi mijin areq. Integre{

lov (3.14)- miavor sferayov, kstananq

∫2π ∫ ( ) 2π n φ(z) 6 dt φ z + ωreit dσ(ω) = (n − 1)! 2π S ∫ 0 = φ (z + ωr) dσ(ω), S

Glux ԼԼԼ. Ps dou ucik tiruyner orte

2π n (n − 1)!

iv miavor sferayi avaln :

 e o r e m 3.1. Ee φ -n plyurisubharmonik  D -um, apa bavakanaa

oqr r -i hamar nra areq kamayakan z ∈ ∈ D ketum i gerazancum nra areqneri mijinic S(z, r) sferayov` ∫ φ(z) 6

(n − 1)! 2π n

H e t a n q 3.1.

Cn

φ (z + ωr) dσ(ω) :

(3.15)

S

D⊂ funkcia 2n

subharmonik monik funkcia :

tiruyum kamayakan plyuri{ irakan o oxakani subhar{

 e o r e m 3.2. Ee φ funkcian plyurisubharmonik  z0

keti rjakayqum, apa nra (n − 1)! S(r) = 2π n

∫ φ (z0 + ωr) dσ(ω) S

mijin areq {|z−z0 | = r} sferayi vra r -ic ao funkcia : A p a c u y c: Iskapes, (3.15)-ic het um , or (n − 1)! S(r) = 2π n

∫

dσ(ω) 2π

∫2π

( ) φ z + ωreit dt,

S

het abar, bavakan  nkatel, or u(ζ) = φ (z + ωζ) subharmonik funkciayi mijin, aysinqn s(r) = 2π

∫2π u(reit )dt

me uyun ao funkcia  r -ic:

§ 15.

Plyurisubharmonik funkcianer

6. Plyurisubharmonik funkcianeri motarkum. Aym apa{ cucenq, or kamayakan plyurisubharmonik funkcia motarkvum  nmanatip, bayc anverj diferenceli funkcianerov:

 e o r e m 3.3. Kamayakan φ funkciayi hamar, or plyu{

risubharmonik  D ⊂ Cn tiruyum, kareli  ka ucel ao ∞ ∪ bac bazmuyunneri Gk , Gk = D hajordakanuyun k=1 nvazo φk ∈ C ∞ (Gk ) plyurisubharmonik funkcianeri ha{ jordakanuyun, or zugamitum  φ -in cankaca z ∈ D ketum: A p a c u y c: Ee φ ≡ −∞ , apa orpes φk kvercnenq

hajordakanuyun: ndhanur depqum ka ucvum  mijinacmamb: Nermu enq φk (z) ≡

−k

φk -n

{ ce−1/(1−|z| ) ,

ee |z| < 1, 0, ee |z| > 1 funkcian, orte c hastatun ntrva  aynpes, or K -i in{ tegral amboj Cn -ov lini havasar meki: Ayd funkcian menq kgtagor enq orpes mijinacno koriz: Ditarkenq K(z) =

∫

φk (z) = Cn

( w) φ z+ K(w)dV (w) k

(3.16)

funkcianer, orte integrum astoren katarvum  miavor gndov, qani or gndic durs K ≡ 0 : Parz , or yuraqanyur φk orova  { } Gk =

z ∈ D : δ(z, ∂D) >

k

bac bazmuyan vra, orte δ -n vklidyan he avoruyunn : Parz ∞ ∪  na , or Gk ⊂ Gk+1 bolor k -eri hamar or Gk = D : k=1

Glux ԼԼԼ. Ps dou ucik tiruyner w

Katarelov z + k grenq het yal tesqov` φk (z) = k

7→ w ∫ 2n

o oxakani oxarinum, (3.16)-

( ) φ (w) K k(w − z) dV (w),

Cn

orteic er um , or φk ∈ C ∞ (Gk ) , orovhet K -n anverj dife{ renceli : Cuyc tanq, or φk -n bavararum  (3.14)-ov artahayt{ vo plyurisubharmonikuyan haytaniin: Iroq, ee z ∈ Gk , ω ∈ Cn r - bavakanin oqr , apa 2π

∫2π

( ) φ z + ωreit dt =

∫ = Cn

   1 ∫2π (  ) w it K(w) φ z + + ωre dt dV (w) >  2π  k ∫ ( w) > K(w)φ z + dV (w) = φk (z) : k Cn

Ayste menq gtvel enq ayn banic, or φ -n plyurisubharmonik  isk K -n` o bacasakan: Ancnelov (3.16)-um b e ayin koordinatneri (tes (4.11)) dV (rζ) = r2n−1 dr dσ(ζ),

orte

ζ ∈ S,

r = |w|

kstananq ∫

∫1 φk (z) =

r

2n−1

K(r)dr

( ) rζ φ z+ dσ(ζ) = k

S

2π n = (n − 1)!

∫1 r2n−1 K(r)S

(r) k

dr,

(3.17)

§ 15.

Plyurisubharmonik funkcianer

(r)

orte S -n irenic nerkayacnum  φ funkciayi areqneri k mijin {w : |w − z| = r/k} sferayi vra: st eorem 3.2-i, k -n aelis φk -n nvazum : norhiv φ funkciayi plyurisubharmonikuyan nra mijin S(r/k) > φ(z) st eorem 3.1-i , qani or 2π n (n − 1)!

∫1

∫1 r

2n−1

∫ r2n−1 K(r)

K(r)dr = ∫0 =

dσ(ζ) = S

K(w)dV (w) = 1, Cn

apa (3.17)-ic het um , or φk (z) > φ(z) kamayakan z ∈ D , sksa in-or k0 hamaric: Myus komic, φ -i kisaanndhatu{ yunic bxum , or cankaca ε > 0 hamar φ(w) − φ(z) < ε bolor w -neri hamar, oronq bavakanaa mot en z -in: Owremn, S(r/k) 6 φ(z) + ε , erb k > k0 aydpisi k -eri hamar (3.17)-ic stanum enq φk (z) < φ(z) + ε : Ayspisov, stacvec, or lim φk (z) = φ(z)

k→∞

bolor

z∈D

hamar:

D i t o  u  y u n 3.5. Plyurisubharmonik funkcianeri

4 -rd

hatkuyunic het um , or eorem 3.3- hakadareli :

7. L ii  . Inpes haytni , C 2 dasi subharmonik funk{ ∂2

cianer bnuagrvum en nranov, or nranc vra ∆ = 4 ∂ζ ∂ ζ̄ Laplasi perator o bacasakan : Bard funkciayi a ancman kanonic het um ` n ∑ ∂ 2 φ(z 0 + ωζ) ∂ 2 φ(z 0 ) = ωi ω̄k : ∂zi ∂ z̄k ∂ζ ∂ ζ̄ i,k=1

Glux ԼԼԼ. Ps dou ucik tiruyner Ayste aj masum a ajaca   rmityan , orovhet (

=

∂2φ ∂zi ∂ z̄k

)

∂2φ = ∂zi ∂ z̄k

, qani or φ -n irakan : Ayn nanakvum  Hz (φ, ω) =

n ∑ ∂ 2 φ(z) ωi ω̄k ∂zi ∂ z̄k

i,k=1

kovum  φ funkciayi L ii  enq het yal haytani:

z

ketum: Ayspisov, menq stanum

 e o r e m 3.4. Orpeszi φ ∈ C 2(D) funkcian lini plyu{

risubharmonik, anhraet  bavarar, or yuraqanyur z ∈ D ketum nra Hz (φ, ω) L ii   lini o bacasakan bolor ω ∈ Cn vektorneri hamar:

Plyurisubharmonik funkcianeric a annacnenq mi kar or das:

S a h m a n u m 3.1.

subharmonik, ee

1.

φ

funkcian kovum  xist plyuri{

φ ∈ C 2 (D) ,

2. yuraqanyur

z∈D

Hz (φ, ω) > 0

ketum nra L ii  

bolor

ω ∈ Cn \ {0}

vektorneri hamar:

 e o r e m 3.5.

Ee φ funkcian plyurisubharmonik  D⊂ tiruyum, isk ψ funkcian irakan, ao, u ucik  ψ ∈ C 2 (D) , apa (ψ ◦ φ) -n plyurisubharmonik  D -um: Cn

A p a c u y c: Skzbic ditarkenq ayn depq, erb φ ∈ C 2 (D) :

Dvar  stugel L ii  i het yal hatkuyun`

Hz (ψ ◦ φ, ω) = ψ ′ ◦ φ(z)Hz (φ, ω) + ψ ′′ ◦ φ(z) |∂φ(ω)|2 ,

§ 15.

Plyurisubharmonik funkcianer

qani or ψ′ > 0 ψ′′ > 0 , apa ditarkvo depqum eoremi pndum apacucva : ndhanur depqum gtvum enq eorem 3.3-ic nra haka{ daric. φ -n motarkum enq oork plyurisubharmonik funkci{ aneri hajordakanuyamb φk ↘ φ : st ver um apacuca i, ψ ◦ φk funkcianer plyurisubharmonik en: Havi a nelov, or ψ -n ao funkcia , stanum enq ψ ◦φk ↘ ψ ◦φ , uremn, ψ ◦φ -n plyurisubharmonik : 8. Invariantuyun biholomorf artapatkerman nkat{ mamb.

 e o r e m 3.6. Biholomorf artapatkerman amanak plyurisubharmonik funkciayi patker plyurisubharmo{ nik : A p a c u y c: Skzbic enadrenq, or plyurisubharmonik u

funkcian patkanum  C 2 dasin: Dicuq ζ = ζ(z) - biholomorf artapatkerum  B -n B1 -i vra z = z(ζ) hakadar arta{ patkerumn : Nanakelov A -ov A=

matric,

∂zj , ∂ζk

[ ] u1 (ζ) = u z(ζ) (

j, k = 1, . . . , n

funkciayi hamar kunenanq

) ( ) H(ζ; u1 )a, ā = H(z; u)Aa, Aa > 0 :

Isk ee u ∈/ C 2 , apa kira elov stacva ardyunq C ∞ dasi plyurisubharmonik funkcianeri nkatmanb, oronq nvazelov gtum en u -in (tes eorem 3.3-), stanum enq pndum ndhanur depqum:

Glux ԼԼԼ. Ps dou ucik tiruyner

9. Plyurisubharmonik funkciayi hetq analitik make{ r uyi vra.

 e o r e m 3.7.

tiruyum plyurisubharmonik φ funkciayi hetq kamayakan m -a ani f : G 7→ Cn , G ⊂ ⊂ Cm analitik maker uyi vra s plyurisubharmonik  Ω = {ζ ∈ G : f (ζ) ∈ D} bac bazmuyunum: D ⊂ Cn

A p a c u y c: Parzuyan hamar sahmana akvenq m = 1 depqov, aysinqn, apacucenq, or φ -i hetq z = f (ζ) analitik kori vra subharmonik funkcia : Skzbic ditarkenq φ ∈ C 2 (D) depq: Kira elov bard funk{ ciayi a ancman kanon u = φ ◦ ψ funkciayi nkatmamb, ksta{ nanq ( ) n ∑ ∂2u ∂ 2 φ ∂fi = ∂zi ∂ z̄k ∂ζ ∂ζ ∂ ζ̄ i,k=1

∂fk ∂ζ

:

Qani or φ -n plyurisubharmonik , st eorem 3.4-i aj masum masnakco   o bacasakan , isk da nanakum  u(ζ) -i sub{ harmonikuyun: ndhanur depq bervum  ditarkva in norhiv eorem 3.3-i nran het o ditouyan:

H e t a n q 3.2. Analitik S maker uyi vra plyurisub{ harmonik funkciayi hetqi hamar marit  maqsimumi skzbunq: §

16. Ow ucik funkcianer

Ow ucik funkcian plyurisubharmonik funkcianeri kar or masnavor depq :

§ 16.

Ow ucik funkcianer

1. Ow ucik funkciayi sahmanum: u(x) < +∞ funkcian

kovum  u ucik (a, b) mijakayqum, ee cankaca x hamar (a, b) -ic tei uni

x′

( ) u λx + (1 − λ)x′ 6 λu(x) + (1 − λ)u(x′ )

keteri (3.18)

anhavasaruyun:

u(x) = u(x1 , . . . , xn ) funkcian kovum  u ucik B ⊂ Rn ti{ ruyum, ee bolor x0 ∈ B keteri b ∈ Rn ( |b| = 1 ) vektorneri hamar u(x0 + tb) funkcian u ucik  st t -i { } Bx0 ,b = t : x0 + tb ∈ B

bac bazmuyan patkano yuraqanyur mijakayqum: Berva sahmanumneric het um , or ee u(x) - u ucik  B -um, apa kam u(x) ≡ −∞ , kam u(x) - verjavor  B -i bolor keterum: 2. Ow ucik funkciayi anndhatuyun: Ee u(x) ̸≡ −∞

funkcian u ucik  B tiruyum, apa na anndhat :

A p a c u y c: Nax apacucenq, or u(x) - ver ic sahmana{

ak  amen mi K ⊂ B kompakti vra: st Hayne{Boreli lemmayi ayd pndum bavakan  apacucel B -in patkano ak u ucik bazmanisteri hamar: Nanakenq x(k) -ov, k = 1, . . . , N , trva

Π bazmanisti gaganer: Kamayakan x ∈ Π ket nerkayacvum  het yal tesqov` x=

∑

tk x(k) ,

orte

tk > 0

16k6N

∑

tk = 1 :

16k6N

Ayd depqum tei uni u(x) 6

∑ 16k6N

tk u(x(k) ),

x∈Π

(3.19)

Glux ԼԼԼ. Ps dou ucik tiruyner

anhavasaruyun: Iroq, N = 1 depqum (3.19)- it : Ena{ drenq, ayn it  (N − 1) -i depqum: gtvelov (3.19)-ic, stanum enq ( u(x) = u

)

∑

tk x(k) + tN x(N )

16k6N −1

) tk (k) x + 6 (1 − tN )u 1 − tN 16k6N −1 ∑ ( ) ( ) + tN u x(N ) 6 tk u x(k) : (

∑

16k6N

(3.19)-ic bxum  orovhet

u(x) -i

sahmana akuyun ver ic

u(x) 6 max

16k6N

{ ( )} u x(k) ,

Π -i

vra,

x∈Π:

Dicuq x0 ∈ B , apacucenq, or u -n anndhat  x0 -um: st apacuca i, u(x) 6 a bavakanaa

oqr |x − x0 | 6 r gndum: |x − x0 | Dicuq xk → x0 , nd orum 1 > εk = → 0 : Kira elov (3.18)-,

r xk − x0 erb λ = εk , x = + x0 x′ = x0 , stanum εk ( ) xk − x0 u(xk ) 6 εk u + x + (1 − εk )u(x0 ) : εk

enq`

Aysteic het um , or [ ] u(xk ) − u(x0 ) 6 εk a − u(x0 ) :

Kira elov (3.18)-, erb kstananq

λ=

, x = xk 1 + εk

εk u(xk ) + u u(x ) 6 1 + εk 1 + εk

(

x′ =

(3.20) x0 − xk + x0 , εk

) xk − x0 +x , εk

§ 16.

orteic

Ow ucik funkcianer

[ ] u(x0 ) − u(xk ) 6 εk a − u(x0 ) ,

in (3.20)-i het miasin apacucum  u -i anndhatuyun ketum:

x0

3. Ow ucik funkciayi hatkuyunner: Ow ucik funkcia{

yi hatkuyunner nman en anndhat plyurisubharmonik funk{ ciayi hatkuyunnerin bxum en mek o oxakani u ucik funkciayi hatkuyunneric it aynpes, inpes plyurisub{ harmonik funkciayi hatkuyunner bxum en mek (kompleqs)

o oxakani subharmonik funkciayi hatkuyunneric: Nenq drancic mi qanis: Ee u(x) funkcian patkanum  C 2 (B) dasin, apa na u u{ cik  B tiruyum ayn miayn ayn depqum, erb

1.

(

) ∑ H(x; u)b, b = j,k

∂2u bj bk ∂xj ∂xk

qa akusayin   B -um drakan : 2. Ee {uα (x)} u ucik funkcianeri ntaniq vasaraa sahmana ak , apa

B -um

lokal ha{

sup uα (x) α

verin paruri s u ucik  B -um: 3. Ee u(z) = u(x, y) funkcian u ucik  G ⊂ apa ayn plyurisubharmonik  ayd tiruyum: A p a c u y c: Dicuq aj = bj + icj , n ∑ j,k=1

∂2u 1∑ aj āk = ∂zj ∂ z̄k j

(

∂2u ∂2u + 2 ∂x2j ∂yj

)

(

R2n

) b2j + c2j +

tiruyum,

Glux ԼԼԼ. Ps dou ucik tiruyner

[ ) ∑ ( ∂2u ∂2u + + (bj bk + cj ck ) + ∂xj ∂xk ∂yj ∂yk j̸=k ] ( ) ∂2u ∂2u − (cj bk − ck bj ) : (3.21) + ∂xj ∂yk ∂xk ∂yj

Nanakelov B = (b, c) aysteic stanum enq (

= (b1 , . . . , bn , c1 , . . . , cn )

C = (c, −b) ,

) 1( ) 1( ) H(z; u)a, ā = H(x, y; u)B, B + H(x, y; u)C, C ,

orteic bxum  pndum: Orpeszi u(z) funkcian lini u ucik B ⊂ Rn tiruyum, an{ hraet  bavarar, or na lini plyurisubharmonik B + iRn ⊂ ⊂ Cn xoovaka tiruyum:

4.

A p a c u y c: Iroq, qani or

(3.21)-ic het um  (

∂u = 0 , j = 1, 2, . . . , n , ∂yj

apa

) 1( ) 1( ) H(z; u)a, ā = H(x; u)b, b + H(x; u)c, c ,

orteic bxum  pndum:

§

17. Ps dou ucik tiruyner

1. Ps dou ucik tiruyi sahmanum: Ps dou ucik ti{

ruyner sahmanvum en plyurisubharmonik funkcianeri mijo{ cov aynpes, inpes u ucik tiruyner` u ucik funkcianeri mijocov: Ps dou ucikuyun handisanum  irakan Rn tara{

uyan mej sahmanva u ucikuyan gaa ari ndhanracum kompleqs Cn tara uyan depqi vra:

§ 17.

Ps dou ucik tiruyner

G tiruy kovum  ps dou ucik, ee − ln ∆G (z) funkcian plyurisubharmonik  G -um: G tiruyum orova − ln ∆G (z) funkcian gtum  (+∞) - tiruyi ∂G ezri bolor verjavor keterum: Owremn, ee G -n ps { dou ucik , apa { } max − ln ∆G (z), |z|2

anndhat funkcian plyurisubharmonik  (+∞) - amenureq ∂G -i vra:

G -um

gtum 

2. Ps dou ucik tiruyneri parzaguyn hatkuyunner:

Ps dou ucik tiruyneri hatman nersi yuraqanyur ka{ pakcva komponent ps dou ucik : A p a c u y c: Dicuq Gα -ner ps dou ucik tiruyner en:

Ayd depqum − ln ∆Gα (z) funkcianer plyurisubharmonik en ( ∩ )◦ Gα -um: Dicuq G -n handisanum  Gα bazmuyan or  α kapakcva komponent: − ln ∆Gα (z) funkcianer lokal hava{ saraa sahmana ak en G -um, orovhet ∆Gα (z) > ∆G (G′ ) > > 0 , erb z ∈ G′ b G : Havi a nelov (3.13)-, stanum enq, or − ln ∆G (z) = sup {− ln ∆Gα (z)} α

funkcian plyurisubharmonik  G -n ps dou ucik tiruy : Ee G ⊂ Cn

G×D ⊂

Cn+m

G -um

(tes

§ 12.4 ):

Ayspisov,

tiruyner ps dou ucik en, apa tiruy nuynpes ps dou ucik : D ⊂ Cm

A p a c u y c: Qani or G × D = (G × Cm )

∩

(Cn × D) ,

Glux ԼԼԼ. Ps dou ucik tiruyner

apa st naxord ardyunqi bavakan  apacucel, or G × Cm Cn × D tiruyner ps dou ucik en Cn+m -um: Isk ayd pndum het um  − ln ∆G (z) = − ln ∆G×Cm (z, w)

a nuyunic, ori hamaayn − ln ∆G×Cm (z, w) funkcian plyuri{ subharmonik  G × Cm -um, aysinqn` (G × Cm ) tiruy ps do{ u ucik : Ps dou ucik tiruyneri ao hajordakanuyan mia{ vorum nuynpes ps dou ucik : A p a c u y c: Dicuq Gα ⊂ Gα+1

G=

∪ α

− ln ∆Gα (z) > − ln ∆Gα+1 (z) → − ln ∆G (z),

kamayakan

G′ b G

Gα :

st (3.12)-i

erb

α→∞

enatiruyum:

3. Anndhatuyan uyl skzbunq: Kasenq, or G ⊂ Cn ti{

ruyi hamar it  anndhatuyan uyl skzbunq, ee G -um trva kamayakan Sα kompakt analitik koreri hajor{ dakanuyan hamar lim Sα = S0

α→∞

lim ∂Sα = T0 b G

α→∞

paymanic het um , or S0 b G : Aknhayt , or C1 -um amen mi tiruyi hamar it  annd{ hatuyan uyl skzbunq: 4. Ps dou ucikuyan haytani 1: Orpeszi G tiruy{

 lini ps dou ucik, anhraet  bavarar, or nra ha{ mar tei unena anndhatuyan uyl skzbunq:

§ 17.

Ps dou ucik tiruyner

L e m m a 3.1. Ee G tiruyi hamar tei  unenum an{ ndhatuyan uyl skzbunq, apa u(z) = − ln R(z) funkci{ an, orte R(z) - G -i Hartogsi a avin  z ketum, plyu{ risubharmonik  G -um: A p a c u y c: Aknhayt , or kamayakan G tiruyi Har{

togsi a avi hamar lim R(z) > R(a) , aynpes or R - kisa{ z→a anndhat  nerq ic, isk u = − ln R - kisaanndhat  ver ic: Mnum  apacucel, or u(z) = − ln R(z) funkciayi hetq kama{ yakan lω = {z ∈ Cn : z = l(ζ) = a + ωζ} ,

a ∈ G,

ω ∈ Cn

(3.22)

kompleqs ui vra subharmonik  ζ = 0 keti rjakayqum: Ee aysinqn, ee lω -n zugahe  zn a ancqin, apa

ω e =e 0,

R|lω =

inf

z ′ ∈∂G∩lω

zn − zn′ ,

, het abar, u|lω = − ln R|lω =

sup

z ′ ∈∂G∩lω

{ } − ln zn − zn′

funkcian subharmonik ` orpes subharmonik funkcianeri ve{ rin paruri: Ayn depqum, erb ωe ̸= e0 , apacuyc katarenq hakaso ena{ druyamb: Ee u|lω = − ln R ◦ l(ζ) = v(ζ) (3.23) funkcian subharmonik  ζ = 0 keti rjakayqum, apa goyu{ yun uni U = {ζ : |ζ| < r} rjan nranum harmonik h funkcia, orn anndhat  U -um, v(ζ) 6 h(ζ) erb ζ ∈ ∂G , bayc in-or ζ0 ∈ U ketum h(ζ0 ) − v(ζ0 ) = inf {h(ζ) − v(ζ)} = −ε < 0 : U

Glux ԼԼԼ. Ps dou ucik tiruyner

Ayste menq havi enq a el, or nerq ic kisaanndhat h − v funkcian kompakti vra hasnum  ir storin ezrin: Aynuhet , g(ζ) = −h(ζ) − ε funkcian, or anndhat  U -um harmonik  U -um, bavararum  het yal paymannerin` g(ζ) < −v(ζ) g(ζ) 6 −v(ζ)

∂U -i

vra, U -i vra,

(3.24)

g(ζ0 ) = −v(ζ0 ) : ( ) l ζ) = (e l(ζ), λ(ζ) , aynpes or e l(ζ) = e a+ω eζ λ(ζ) = an + ωn ζ : Hamaayn Hartogsi a avi sahmanman, goyuyun uni b = (eb, bn ) ∈ ∂G aynpisi ket, or eb = el(ζ0 ) |bn − −λ(ζ0 )| = R◦l(ζ0 ) : Ka ucenq U -um holomorf G = g+ig∗ funkcia ( ) aynpes, or G(ζ) -n hamnkni ln bn − λ(ζ0 ) areqneric or  meki het: Da hnaravor , orovhet ln bn − λ(ζ0 ) = −v(ζ0 ) = g(ζ0 )

Dicuq (3.22)-um

st (3.23)-i (3.24)-i: Aym ditarkenq kompakt analitik koreri

} { St = z ∈ Cn : ze = e l(ζ), zn = λ(ζ) + teG(ζ) , ζ ∈ U

ntaniq: Kamayakan z ∈ St keti hamar mi komic` myus komic st (3.24)-i erkrord anhavasaruyan`

ze = e l(ζ) ,

zn − λ(ζ) = teg(ζ) 6 te−u(ζ) = tR ◦ l(ζ) :

Aysteic Hartogsi a avi sahmanumic het um , or St ⊂ G , erb t < 1 : norhiv (3.24)-i a ajin anhavasaruyan, nman  ov stanum enq` St ⊂ ∂G , erb 0 6 t 6 1 : Aknhayt , or St → S1 , erb t → 1 : Myus komic, S1 - parunakum  b ∈ ∂G ket, orovhet ζ = ζ0 ketum unenq ze = e l(ζ0 ) = eb zn = λ(ζ0 ) + teG(ζ0 ) = = bn (hienq, or ln G(ζ0 ) = b0 − λ(ζ0 ) ): Stacanq hakasuyun ayn paymani het, or G -n bavararum  anndhatuyan uyl skzbunqin:

§ 17.

Ps dou ucik tiruyner

A n h r a  e  t u  y u n: Enadrenq haka ak, G -n i bavararum anndhatuyan uyl skzbunqin: Ayd depqum go{ yuyun uni Sk kompakt analitik maker uyneri hajordaka{ nuyun, or Sk → S , ∂Sk → ∂S , nd orum S k , ∂S ∈ G , isk S - parunakum  or  a ∈ ∂G ket: Owremn, − ln ∆G (z) funkcian plyurisubharmonik  G -um: st maqsimumi skzbunqi plyuri{ subharmonik funkcianeri hamar sup(− ln ∆G ) 6 sup(− ln ∆G ) < c < ∞, (3.25) Sk

∂Sk

orte c -n kaxva  k -ic, qani or ∂Sk -eri miavorum kompakt  ov patkanum  G -in: Myus komic, goyuyun uni z k ∈ Sk keteri hajordakanuyun, or zugamitum  a ∈ ∂G ketin, − ln ∆G (z k ) → ∞ , in hakasum  (3.25)-in: B a v a r a r u  y u n: Petq  apacucel, or − ln ∆G (z) funkcian plyurisubharmonik  G -um: Nanakenq lω -ov ζ 7→ z + ωζ kompleqs ui, orn ancnum  z ketov ω vektori uuyamb, dicuq Rω (z) = ∆a,G (z) : Akn{ hayt , or ∆G (z) = inf ∆ω,G (z), ω

orte storin ezr vercva  st bolor ω ∈ Cn , |ω| = 1 : z 7→ Cz ptuyti mijocov, orte C -n unitar perator , ω vektori uuyun kareli  tanel zn a ancqi uuyan, aynpes or Rω (z) -n kancni G tiruyi Hartogsi a avin: Qani or ayd ptuyt pahpanum  | vklidyan he avoruyun, | ps { dou ucikuyun, | L -u ucikuyun, apa st lemma 3.1-i ka{ reli  pndel, or − ln Rω (z) - plyurisubharmonik  G -um: Ayd depqum ( ) − ln ∆G (z) = sup − ln ∆ω,G (z) (3.26) ω

funkcian plyurisubharmonik  G -um orpes plyurisubharmo{ nik funkcianeri verin paruri:

Glux ԼԼԼ. Ps dou ucik tiruyner

Orpeszi G ⊂ Cn tiruy lini ps dou ucik, anhraet  bavarar, or − ln ∆ω,G (z) funkcian lini plyurisubharmonik G -um bolor ω -eri (|ω| = 1) depqum: 5. Ps dou ucikuyan haytani

11:

A p a c u y c: Ee G -n ps dou ucik , apa, inpes het um  ps dou ucikuyan haytani I-i bavararuyan apacuycic, − ln ∆ω,G (z) funkcian plyurisubharmonik : Haka ak, ee − ln ∆ω,G (z) funkcian plyurisubharmonik , apa hamaayn (3.26)-i, − ln ∆G (z) - s plyurisubharmonik , isk da nanakum , or G -n ps dou ucik : 6. Ps dou ucikuyan haytani

plyurisubharmonik funkcia 

111:

Dicuq V (z) -

{ } G = z : V (z) < 0, z ∈ U (G)

bac bazmuyan U (G) rjakayqum: Ayd depqum G -i yura{ qanyur kapakcva komponent ps dou ucik : A p a c u y c: Nkatenq, or G bazmuyun bac , orovhet

V (z) - kisaanndhat  ver ic U (G) -um: Nax enadrenq e G -n sahmana ak bazmuyun , ayd depqum G b U (G) : Skzbic ditarkenq anndhat V (z) -i depq: Inpes ps dou ucikuyan

haytanii anhraetuyan apacuyci amanak, miayn e o{ xarinelov − ln ∆G (z) - V (z) -ov, cuyc  trvum, or G bac bazmu{ yan hamar it  anndhatuyan uyl skzbunq: Owremn` G -i yuraqanyur kapakcva komponent ps dou ucik tiruy : Aym azatvenq ayn enadruyunic, e V (z) - anndhat : st eorem 3.3-i, goyuyun uni anndhat Vα (z) funkcianeri hajordakanuyun, oronq plyurisubharmonik en G′ bac bazmu{ yan vra, G′ b G b U (G) , zugamitum en V (z) -in: Het abar { } Gα = z : Vα (z) < 0, z ∈ G′ ,

α = 1, 2, . . .

§ 17.

Ps dou ucik tiruyner ∪

bac bazmuyunneri hajordakanuyun aum  Gα = G : α st apacuca i, Gα -i yuraqanyur kapakcva komponent ps dou ucik tiruy : Hamaayn § 14.2 -i eoremi, G -i yu{ raqanyur kapakcva komponent ps dou ucik : Aym enadrenq G bazmuyun sahmana ak : Ditarkenq ∩ sahmana ak bac bazmuyunneri GR = G B(0, R) , R = 1, 2, . . . hajordakanuyun: Parz , or { } GR = z : VR (z) < 0, z ∈ U (G) ,

orte

GR b U (G),

{ } VR (z) = max V (z), |z|2 − R2

funkcian plyurisubharmonik  U (G) -um: st apacuca i, GR bazmuyan amen mi kapakcva komponent ps dou ucik ti{ ∪ ruy : Myus komic, GR ⊂ GR+1 GR = G : st § 14.2 -i, R>0 G bazmuyan yuraqanyur kapakcva komponent ps dou u{ cik : 7. Ps dou ucikuyan haytani 1Մ: Orpeszi G tiruy{  lini ps dou ucik, anhraet  bavarar, or G -um go{ yuyun unena plyurisubharmonik V (z) funkcia, or g{ tum  +∞ amenureq ∂G -i vra:

A p a c u y c: Anhraetuyun bxum  ps dou uciku{ yan sahmanumic ayn astic, or − ln ∆G (z) plyurisubharmo{ nik funkcian gtum  +∞ - amenureq ∂G -i vra: Apacucenq bavararuyun: Qani or V (z) - gtum  +∞ amenureq ∂G -i vra, apa Gα = {z : V (z) − α < 0, z ∈ G} ,

α = 1, 2, . . . ,

Glux ԼԼԼ. Ps dou ucik tiruyner

bac bazmuyunneri hajordakanuyun tva  het yal hat{ kuyunnerov` Gα ⊂ Gα+1 b G,

∪

Gα = G :

α

Qani or Gα b G V (z) − α funkcian plyurisubharmonik  G -um, apa Gα bac bazmuyan yuraqanyur kapakcva kom{ ponent handisanum  ps dou ucik tiruy: st § 14.2 -i, G -n ps dou ucik : 8. Ps dou ucikuyan haytani Մ:

 e o r e m 3.8. Dicuq G -n ps dou ucik tiruy  V (z) funkcian plyurisubharmonik  G -um: Het yal bac bazmu{ yan` G′ = {z ∈ G : V (z) < 0}

amen mi kapakcva komponent ps dou ucik tiruy : A p a c u y c: st § 14.7 -i, goyuyun uni plyurisubharmonik V ∗ (z) funkcia, or {z ∈ G : V ∗ (z) < 0} b G, Vα (z) = max {V ∗ (z) − α, V (z)} en G -um

G -um

aynpisi

α = 1, 2, . . . :

funkcianer plyurisubharmonik

Gα = {z ∈ G : Vα (z) < 0} ,

α = 1, 2, . . .

bac bazmuyunneri hajordakanuyun bavararum  Gα b G,

paymannerin: st ps dou ucik :

Gα ⊂ Gα+1

§ 14.2 -i

∪

Gα = G′

α

ps dou ucikuyan haytanii, G -n

§ 17.

Ps dou ucik tiruyner

9. Xist ps dou ucik tiruy: Ee tiruyi ezr erku angam oork , apa ps dou ucikuyun kareli  nkaragrel o e spa i, inpes § 14.1 -um, ayl oroi funkciayi mijocov: ρ -n kanvanenq oroi funkcia D ⊂ Cn tiruyi hamar, ee ∂D ezri in-or Ω rjakayqum na patkanum  C 2 dasin, ayd rjakayqum ∩ D Ω = {z ∈ Ω : ρ(z) < 0} , (3.27) grad ρ(z) ̸= 0 , erb z ∈ ∂D :

S a h m a n u m 3.2.

ezrov D ⊂ Cn tiruy kovum  ps dou ucik, ee na uni oroi ρ funkcia, ori L ii   Hz (ρ, ω) > 0 bolor z ∈ ∂D ω ∈ Tzc (∂D) hamar, kovum  xist ps dou ucik, ee na sahmana ak  Hz (ρ, ω) > 0 , erb z ∈ ∂D , ω ∈ Tzc (∂D) , ω ̸= 0 : C2

 r i n a k 3.1. Ditarkenq

B n = {z ∈ Cn : |z| < 1} gund: n ∑ Orpes oroi funkcia karo  a ayel ρ(z) = zk z̄k − 1 funk{ n ∑

k=1

cian, nra L ii   Hz (ρ, ω) = ωk ω̄k = k=1 Owremn, gund xist ps dou ucik tiruy :

|ω|2

> 0,

erb

ω ̸= 0 :

Cuyc tanq, or 3.2 sahmanman mej oroi funkciayi ntru{ yun der i xaum: Ayd npatakov nax apacucenq

L e m m a 3.2.

Ee G ⊂ Rn tiruyum orova en φ, ψ ∈ ∈ funkcianer, nd orum grad φ(z) ̸= 0 ψ = 0 ame{ nureq, orte φ = 0 , apa goyuyun uni aynpisi h ∈ C k−1 (G) funkcia, or ψ = hφ G -um: C k (G)

A p a c u y c: Qani or lemmayi pndum krum  lokal bnuy, apa kareli  hamarel, or G -n skzbnaketi rjakayq , orte φ(x) = 0 havasarum hamareq  xn = φ1 (e x) -in: Katare{ lov xe 7→ xe , xn 7→ xn − φ1 (ex) o oxakani oxarinum, harc

Glux ԼԼԼ. Ps dou ucik tiruyner khangecnenq ayn depqin, erb parz a nuyunic` ∫1 ψ(x) = xn

φ(x) = xn

∂ ψ(e x, txn ) dt, ∂xn

ψ(e x, 0) = 0 :

Het yal

x ∈ G,

er um , or orpes h kareli  vercnel ayste masnakco inte{ gral, or aknhaytoren patkanum  C k−1 (G) -in: Aym enadrenq, or D tiruyn uni φ ψ oroi funkcianer: st lemmayi, kamayakan ezrayin a keti Ua rjakayqum goyu{ yun uni aynpisi ha ∈ C 1 (Ua )∩funkcia, or ψ = hφ : Qani or φ -n, ψ -n bacasakan en Ua D -um drakan en Ua \ D -um, Ua \ D bazmuyan vra h > 0 : Bayc h ̸= 0 na ∂D -i vra, orovhet , st nuyn lemmayi, φ = h1 ψ , orte h1 = 1/h ∈ C 1 (Ua ) , uremn h > 0 amenureq Ua -um: Myus komic, dvar  stugel, or Ha (ψ, ω) = h(a)Ha (φ, ω), erb ω ∈ Tac (∂D), aynpes or φ ψ funkcianeri L ii  eri hetqer Tac (∂D) -i vra tarbervum en drakan artadriov: Nenq ps dou ucikuyan plyurisubharmonikuyan kap:

 e o r e m 3.9.

tiruy xist ps dou ucik  ayn miayn ayn depqum, ee na uni xist plyurisubharmonik oroi funkcia: D ⊂ Cn

A p a c u y c: Paymani bavararuyun aknhayt , apa{ cucenq anhraetuyun: Dicuq ρ -n (3.27) D tiruy or  oroi funkcia : Ee Ω ert bavakanin ne , apa kamaya{ kan k hastatuni hamar oroi  na ψ = ρ + kρ2 funkcian: Anmijakan havum cuyc  talis, or Hz (ψ, ω) = Hz (ρ, ω) + 2k |∂ρ(ω)|2 ,

(3.28)

§ 17.

Ps dou ucik tiruyner

n ∂ρ ∑

ωk : Bavakan  apacucel, or orte ∂ρ(ω) = k=1 ∂zk drakan 

Hz (ψ, ω) -n

E = {(z, ω) : z ∈ ∂D, ω ∈ Cn , |ω| = 1}

kompakt bazmuyan vra: Nanakenq E0 = {(z, ω) ∈ E : Hz (ρ, ω) 6 0} :

Ee E0 -n datark , apa kvercnenq k = 0 , haka ak depqum ntrum enq aynpisi M hastatun, or Hz (ρ, ω) > −M bolor (z, ω) ∈ E0 keteri hamar: Hamaayn xist ps dou ucik tiruyi sahmanman, Hz (ρ, ω) > 0 erb z ∈ ∂D ω ∈ Tzc (∂D) , aysinqn` ∂ρ(ω) = 0 , erb ω ̸= 0 , het abar ∂ρ(ω) ̸= 0 E0 -i vra: Kompaktuan norhiv goyuyun uni m > 0 iv aynpisin, or M , (3.28)-ic stanum |∂ρ(ω)| > m erb ω ∈ E0 : ntrelov k > 2m2 enq, or Hz (ψ, ω) > −M + 2km2 > 0,

z ∈ E0 ,

isk E \ E0 -i vra Hz (ψ, ω)  i drakan linel aknhayt : Annd{ hatuyan nkata umneric parz , or ee Ω -n bavakanin ne , apa Hz (ψ, ω) > 0 erb ω ̸= 0 o miayn ∂D -i vra, ayl na bolor z ∈ Ω keteri hamar, isk da l nanakum  ψ funkciayi xist plyurisubharmonikuyun: 10. Invariantuyun biholomorf artapatkerumneri nkatmamb: Hetagayum mez anhraet  linelu het yal lemman`

L e m m a 3.3. Dicuq biholomorf ζ = ζ(z) artapatkerum

tiruy  a oxum  G1 tiruyi u(z) funkcian [ ] gtum  +∞ amenureq ∂G -i vra: Ayd depqum u z(ζ) funkcian, orte z = z(ζ) -n hakadar artapatkerumn , gtum  +∞ amenureq ∂G1 -i vra:

G

Glux ԼԼԼ. Ps dou ucik tiruyner

A p a c u y c: Kamayakan irakan M vi depqum {z : u(z) < M, z ∈ G} b G :

Qani or biholomorf artapatkerum homeomorf , apa { [ ] } ζ : u z(ζ) < M, ζ ∈ G1 b G1 :

Biholomorf ζ = ζ(z) artapatkerman amanak ps do{ u ucik G tiruyi G1 patker s ps dou ucik : A p a c u y c: Qani or G -n ps dou ucik , apa goyuyun

uni

G -um

plyurisubharmonik V (z) funkcia, or gtum  + [ ] +∞ amenureq ∂G -i vra: Ayd depqum V z(ζ) funkcian, orte z = z(ζ) -n hakadar artapatkerumn , plyurisubharmonik  G1 -um , st naxord lemmayi, gtum  +∞ amenureq ∂G1 -i vra: st § 14.7 -i, G1 - ps dou ucik : 11. Ps dou ucik tiruyneri hatuyner: Ps dou u{

cik G tiruyi g hatuy 2k -a ani analitik F haru{ yamb ps dou ucik tiruy  Ck -um: A p a c u y c: Qani

or biholomorf artapatkerum pahpanum  ps dou ucikuyun, karo enq hamarel, or F haruyun trvum  zn−k = · · · = zn = 0 havasarumnerov: Ps dou ucik G tiruyi hamar goyuyun uni G -um plyuri{ subharmonik V (z) funkcia, or gtum  +∞ amenureq ∂G -i vra: Ayd depqum V (z1 , . . . , zk , 0, . . . , 0) funkcian plyurisub{ harmonik  g -um gtum  +∞ amenureq ∂G -i vra: Owremn, g -i amen mi kapakcva komponent ps dou ucik :

§ 17.

Ps dou ucik tiruyner

12. Hartogsi ps dou ucik tiruy: Hartogsi ps do{

u ucik lriv

G

tiruy kareli  nkaragrel het yal  ov` G = {z : |zn | < R(e z ), ze ∈ B} ,

orte B -n tiruy  Cn−1 tara uyan mej, isk − ln R(ez ) funk{ cian nerq ic kisaanndhat  B -um:  e o r e m 3.10. Hartogsi lriv G tiruy ps dou ucik  ayn

miayn ayn depqum, erb B -n ps dou ucik tiruy  − ln R(e z ) funkcian plyurisubharmonik  B -um:

A p a c u y c: Dicuq G -n ps dou ucik tiruy : Ayd dep{

qum B -n orpes G -i hatuy analitik zn = 0 haruyamb s ps dou ucik  (tes § 14.11 -): st § 14.5 -i, − ln ∆a,G (z) funkcian plyurisubharmonik  G -um: Vercnelov a = (0, . . . , 0, 1) nkatelov, or − ln ∆a,G (e z , 0) = − ln R(e z ),

stanum enq, or − ln R(ez ) funkcian plyurisubharmonik : Hakadar, enadrenq, or B -n ps dou ucik tiruy  − − ln R(e z ) funkcian plyurisubharmonik  B -um: st § 14.2 -i, B × C tiruy ps dou ucik  ln |zn | − ln R(e z ) funkcian plyurisubharmonik  B × C -um: Parz , or { } G = z : ln |zn | − ln R(e z ) < 0, z ∈ B × C1 :

st eorem 3.8-i,

G

tiruy ps dou ucik :

13. kayi eorem: Behenke{Zomeri eoremic het um , or holomorfuyan tiruyi hamar tei  unenum anndhatuyan uyl skzbunq: Aysteic ps dou ucikuyan I haytaniic (tes § 14.4 ) bxum , or holomorfuyan tiruy ps dou u{ cik : Hakadar pndum kazmum  L ii haytni himnaxndri bovandakuyun, orn a ajarkvel  de 1911 .: Ayd himnaxndri lu um tvel  kan 1942 vakanin:

Glux ԼԼԼ. Ps dou ucik tiruyner

 e o r e m 3.11 (ka). Amen mi ps

lomorfuyan tiruy :

dou ucik tiruy ho{

Apacuyc bavakanin bard  pahanjum  bazmaa komp{ leqs analizi lracuci mijocneri nergravum (rinak, Kuzeni xndri lu eliuyun), oronq suyn dasagrqum en ditarkvum: Aynpes or ayd eoremi apacuyc enq beri: Stor varkenq holomorfuyan tiruy bnuagro pay{ manner:

 e o r e m 3.12. Het

yal paymanner hamareq en.

1. G -n holomorfuyan tiruy , aysinqn, goyuyun uni funkcia O(G) -ic, or i arunakvum aveli layn ti{ ruyi mej (tes § 8 -); 2. G -n holomorf u ucik , aysinqn, K b G paymanic he{ t um , or K -i holomorf u ucik aan Kb O b G (tes § 9 -); 3. G -i hamar tei  unenum anndhatuyan uyl skz{ bunq, aysinqn, G -um trva kamayakan Sα kompakt analitik koreri hajordakanuyan hamar lim Sα = S0

α→∞

lim ∂Sα = T0 b G

α→∞

paymanic het um , or S0 b G (tes § 14.3 -); 4. G -n ps dou ucik , aysinqn, − ln ∆G (z) funkcian plyurisubharmonik  G -um (tes § 14.1 -):

§ 18.

§

Ow ucik tiruyner

18. Ow ucik tiruyner

Ow ucik tiruyner sahmanvum en ow ucik funkcianeri mi{ jocov it aynpes, inpes ps dou ucik tiruyner` plyuri{ subharmonik funkcianeri mijocov: Aynpes or u ucik ps { dou ucik tiruyneri mij ka xor nmanuyun: Stor menq karadrenq u ucik tiruyneri mi qani hatkuyun, oronq nd{ g um en nranc nmanuyun ps dou ucik tiruyneri het: 1. Ow ucik tiruyi sahmanum: B ⊂ Rn tiruy kovum

 u ucik, ee {

x′ ∈ B

x′′ ∈ B

paymanic het um , or

} x : x = tx′ + (1 − t)x′′ , 0 6 t 6 1 ⊂ B :

Inpes haytni , tiruy u ucik  ayn miayn ayn amanak, erb nra kamayakan ezrayin ketum goyuyun uni henman har{ uyun: Aysteic bxum . orpeszi {B tiruy lini }u ucik, anhra{ et  bavarar, or na lini x : a(x − x0 ) > 0 kisatara u{ yunneri hatman ners, aysinqn, ( B=

∩ { x0 ∈∂B

{

}

x : a(x − x0 ) > 0

}

)◦ :

Ayste x : a(x − x0 ) = 0 bazmuyun handisanum  ketum B tiruyi henman haruyun:

x0 ∈ ∂B

2. Ow ucikuyan a ajin payman: Orpeszi B tiruy

lini u ucik, anhraet  bavarar, or nra hamar tei unena het yal uyl anndhatuyan skzbunq. ee Sα -n

Glux ԼԼԼ. Ps dou ucik tiruyner

( α = 1, 2, . . . , )∪uagi mijakayqeri aynpisi hajordakanu{ yun , or Sα ∂Sα b B , lim Sα = S0 sahmana ak , lim ∂Sα = = T0 b B , apa S0 b B : A n h r a  e  t u  y u n: Dicuq B -n u ucik tiruy :

Apacucenq, or

∆B (x) > ∆B (∂Sα ),

(3.29)

x ∈ Sα :

Ee (3.29)- tei unenar, kgtnver aynpisi mi x0 ∈ Sα ket, ori hamar ∆B (x0 ) < ∆B (∂Sα ) : (3.30) Nanakenq x′ -ov or  ket ∂B -ic, ori he avoruyun x0 -ic ha{ vasar  ∆B (x0 ) : x′ -ov tanenq Sα -in zugahe L ui: Dicuq ∂Sα = {aα , bα } : st ka ucman, L ui vra kgtnven aynpisi a′α b′α keter, or |aα − a′α | = ∆B (x0 ) , |bα − b′α | = ∆B (x0 ) : Aysteic (3.30)-ic het um , or a′α ∈ B b′α ∈ B : Qani or x0 ket gtnvum  aα -i bα -i mij , apa x′ ket s gtnvum  a′α -i b′α keteri mij : Da hnaravor , orovhet tiruy u ucik : Stacva

hakasuyun apacucum  (3.29) anhavasaruyun: Ancnelov ayd anhavasaruyan mej sahmani, erb α → ∞ , gtvelov ∆B (x) funkciayi anndhatuyunic, S0 u T0 bazmuyunneri sahmana akuyunic, stanum enq ∆B (S0 ) > ∆B (T0 ) : norhiv T0 b B paymani, aysteic bxum , or ∆B (S0 ) > 0 , in S0 -i sahmana akuyan het miasin talis  S0 b B : Isk da na{ nakum , or B tiruyi hamar tei  unenum anndhatuyan uyl skzbunq: B a v a r a r u  y u n. Dicuq x′ ∈ B x′′ ∈ B : Ayd keter miacnenq ktor a ktor oork korov` {x : x = x(t), 0 6 t 6 1} ,

x′ = x(0), x′′ = x(1),

or liovin nka  B tiruyi mej: Qani or x′ - ket , bavakanaa

oqr t -eri hamar x′ = x(0)

B -i nerqin x(t) keter

§ 18.

Ow ucik tiruyner

miacno bolor uagi hatva ner patkanum en B -in: st anndhatuyan uyl skzbunqi, ayd hatva ner patkanum en B -in ∀ t ∈ [0, 1] depqum, masnavorapes, t = 1 -i depqum: Owremn, B -n u ucik tiruy : 3. Ow ucikuyan erkrord payman: Orpeszi B tiruy

lini u ucik, anhraet  bavarar, or − ln ∆B (x) funk{ cian lini u ucik B -um:

B -n u ucik tiruy : A n h r a  e  t u  y u n: Dicuq { }

Yuraqanyur x ∈ ∂B ketov tanenq x : a(x − x ) = 0 henman haruyun ( |a| = 1 ): Inpes haytni , a(x − x0 ) iv havasar  x ∈ ∂B keti he avoruyan min ayd haruyun: Owremn` ∆B (x) = inf

x0 ∈∂B

{

} a(x − x0 )

, het abar, − ln ∆B (x) = sup

x0 ∈∂B

Myus komic, (

− ln a(x − x0 )

{ } − ln a(x − x0 ) :

(3.31)

funkcian u ucik  B -um, orovhet

) ∑ aj ak bj bk (ab)2 H(x; − ln a(x − x0 ))b, b = = >0: [a(x − x0 )]2 [a(x − x0 )]2 j,k

Aynuhet , parz , or {− ln a(x − x0 ), x0 ∈ ∂B} funkcianeri ntaniq havasaraa sahmana ak  ver ic amen mi B ′ b B enatiruyi vra: Ev, uremn, (3.31)-ic het um , or − ln ∆B (x) funkcian u ucik  B -um: B a v a r a r u  y u n: Dicuq − ln ∆B (x) funk{ cian u ucik  B -um: Apacucenq, or B tiruyi hamar tei  unenum anndhatuyan uyl skzbunq: Dicuq Sα -n ( α = = 1, 2, . . . , ) uagi mijakayqeri aynpisi hajordakanuyun

Glux ԼԼԼ. Ps dou ucik tiruyner

∪ Sα ∂Sα b B , lim Sα = S0 sahmana ak , lim ∂Sα = = T0 b B : Ow ucik − ln ∆B (x) funkciayi hamar marit 

, or

maqsimumi skzbunq`

− ln ∆B (x) 6 sup {− ln ∆B (x)} ,

x ∈ Sα ,

x∈∂Sα

or hamareq  (3.29)-in: Inpes menq tesanq ow ucikuyan a ajin paymani anhraetuyan apacuyci amanak, (3.29)-ic het um r anndhatuyan uyl skzbunq B tiruyi hamar: st ow ucikuyan a ajin paymani, B -n u ucik : 4. Ow ucik tiruyneri hatkuyunner: Stor varka

hatkuyunnern apacucvum en it aynpes, inpes ps do{ u ucik tiruyneri depqum, hamapatasxan parzecumnerov. 1. Ow ucik tiruyneri hatman ners s u ucik : 2. Ow ucik tiruyneri ao hajordakanuyan gumar s u u{ cik : 3. Orpeszi B tiruy lini u ucik, anhraet  bavarar, or B -um goyuyun unena u ucik funkcia, or amenureq ∂B -i vra gtum  +∞ : 4. Ee V (x) funkcian u ucik  U (B) -um, apa { } B = x : V (x) < 0, x ∈ U (B)

tiruy u ucik : 5. Ee B -n u ucik tiruy  V (x) - u ucik  B -um, apa B ′ = {x : V (x) < 0, x ∈ B} tiruy s u ucik : 6. Orpeszi B tiruy lini u ucik, anhraet  bavarar, or na lini xist u ucik tiruyneri ao hajordakanuyan gumar` { } Bα = x : Vα (x) < 0, x ∈ U (B α ) ,

α = 1, 2, . . . :

§ 19.

Holomorfuyan aan

Ayste Bα b Bα+1 b B , nd orum Vα -ner anverj diferenceli en U (B) -um: Xist u ucikuyun sahmanvum  xist ps do{ u ucikuyan nmanuyamb: Aym berenq u ucik tiruyneri veraberyal eoremneri ki{ ra uyan rinak: 5. Ps dou ucik xoovaka tiruyner: Orpeszi xo{

ovaka TB tiruy lini ps dou ucik, anhraet  bavarar, or nra B himq lini u ucik:

Ays pndum het um  bana ic:

§

§ 13.3 -ic

− ln ∆T( (z) = − ln ∆B (x)

19. Holomorfuyan aan

1. Holomorfuyan aani sahmanum: Kasenq, or G ti{

ruy D tiruyi holomorf ndlaynum , ee G ⊃ D D -um yuraqanyur holomorf funkcia analitikoren arunakvum  G -i mej: Ver bazmaiv rinakner vkayum en, or Cn , n > 1 ta{ ra uyan mej kan tiruyner, oronc hamar goyuyun uni o trivial holomorf ndlaynum: Ayd ast kapva  tiruyi erk{ raa akan bnuyi het kaxva  or  konkret funkciayic, or holomorf  tiruyum: A ajanum  bnakan xndir. ka ucel amename holomorf ndlaynum:

S a h m a n u m 3.3.

e D

morfuyan aan, ee`

tiruy kovum 

1. amen mi funkcia, or holomorf  runakvum  De -i mej,

D -um,

D

tiruyi holo{

analitikoren a{

Glux ԼԼԼ. Ps dou ucik tiruyner

e 2. yuraqanyur z 0 ∈ De keti hamar goyuyun uni f0 ∈ O(D) ( 0 ) e polidiski vra i a{ funkcia, ori hetq U z , ρ(z , ∂ D) runakvum analitikoren o mi U (z 0 , R) polidisk, orte e : R > ρ(z 0 , ∂ D)

Ays sahmanumic het um , or C1 haruyan vra amen mi tiruy hamnknum  ir holomorfuyan aani het: Ayd pat{ a ov holomorfuyan aani gaa ar C1 -um o mi nana{ kuyun uni: D i t o  u  y u n 3.6. Holomorfuyan aan karo  goyuyun unenal orpes tiruy Cn -um: Gor  nranum , or oro funkcianer arunakman nacqum da num en bazmareq hark  linum ditarkel yuavorva tiruyner, oronq iman{ yan maker uyneri tara akan nmanakner en: Ayd tiruyner kovum en veradrman tiruyner Cn -i vra: Ee ditarkenq na veradrman tiruyner, apa De holomorfuyan aan mit goyuyun uni: 2. aani hatkuyunner:

 e o r e m 3.13.

tiruyi De holomorfuyan aan holomorfuyan tiruy : D

A p a c u y c: Bavakan ( apacucel, or De -n holomorf ) e e

u ucik : Dicuq K b D ρ K, ∂ D = r : st miaamanakya arunakman veraberyal lemma 2.1-i, amen mi f ∈ O(D) funkcia analitikoren arunakvum  U (z, r) polidiski mej, ori kentro{ n kamayakan z ∈ Kb O(D) ket : Qani or st paymani goyuyun ( ) e e > r uni D -ic durs arunakvo holomorf funkcia, ρ z, ∂ D ( ) , uremn, ρ Kb O(D) , ∂ De > r : Myus komic, )ayd he avoruyun ( ( ) b i karo r -ic me linel, usti ρ KO(D) , ∂ De = ρ K, ∂ De , inic het um , or De -n holomorf u ucik :

§ 19.

Holomorfuyan aan

H e t a n q 3.3. Ee D tiruyn uni holomorfuyan aan, apa verjins D -n parunako holomorfuyan ame{ na oqr tiruyn  (aysinqn, D -n parunako bolor holomor{ fuyan tiruyneri hatum):  e o r e m 3.14. Ee G -n D tiruyi holomorf ndlaynum , apa amen mi f ∈ O(D) funkciayi analitik arunakuyu{ n G\D -um karo  ndunel miayn ayn areqner, oronq f - ndunum  D -um: A p a c u y c: Apacucenq hakaso enadruyamb. ena{

drenq, or or  f ∈ O(D) funkcia G\D -um ndunum  w0 areq, or i ndunum D -um: Ayd depqum g(z) =

f (z) − w0

funkcian holomorf  D -um, bayc analitikoren i arunakvum G -i mej, in hakasum  holomorf ndlaynman sahmanman:

H e t a n q 3.4.

sahmana ak tiruyi G holomorf ndlaynum s sahmana ak : D

A p a c u y c: st eorem 3.14-i, fk (z) = zk koordinatakan

funkcianer aysinqn`

G -um

ndunum en nuyn areqner, in or

sup |zk | = sup |zk |, z∈G

D -um,

k = 1, . . . , n :

z∈D

Qani or D -n sahmana ak , apa ays havasaruyunneri aj komer verjavor en, uremn, verjavor en ax komer s, isk da nanakum , or G -n sahmana ak :

L e m m a 3.4. Ee D -n holomorfuyan tiruy , apa nra

r -ndlaynman

Dr = {z ∈ D : ρ(z, ∂D) > r}

Glux ԼԼԼ. Ps dou ucik tiruyner

kamayakan ∆ kapakcva komponent s holomorfuyan ti{ ruy : A p a c u y c: Dicuq K b ∆

z ∈ K

aynpisi

ζ ∈ ∂D mi z ′ ∈ ∂∆

keteri hamar ket, or

ρ(K, ∂∆) = ρ : [z, ζ] hatva i

Kamayakan vra kgtnvi

ρ(z, ζ) = ρ(z, z ′ ) + ρ(z ′ , ζ) > ρ + r :

Het abar, ρ(K, ∂D) > ρ + r , qani or ruy , apa st eorem 2.9-i

D -n

holomorfuyan ti{

b O(D) , ∂D) > ρ + r : ρ(K

(3.32)

Petq  apacucel, or ρ(Kb O(∆) , ∂∆) > ρ , aysinqn, or ρ(z 0 , z ′ ) > ρ kamayakan z 0 ∈ Kb O(∆) z ′ ∈ ∂∆ keteri hamar: Ayn banic, or b O(∆) ⊂ K b O(D) , het abar, z 0 ∈ K b O(D) , ∆ ⊂ D bxum , or K hamaayn (3.32)-i, ρ + r 6 ρ(z 0 , ∂D) 6 ρ(z 0 , z ′ ) + ρ(z ′ , ∂D) = ρ(z 0 , z ′ ) + r :

Menq havi enq a el, or ρ(z ′ , ∂D) = r , qani or z ′ ∈ ∂∆ : Aysteic l bxum , or ρ(z 0 , z ′ ) > ρ :

 e o r e m 3.15. Ee D ⊂ G

patasxanabar

e ⊂G e D

e D

e G

ayd tiruynern unen hama{ holomorfuyan aanner, apa

e ∂ G) e > ρ(∂D, ∂G) : ρ(∂ D,

A p a c u y c: Qani or O(G) ⊂ O(D) , apa amen mi f ∈

e -i mej, aysinqn, ∈ O(G) funkcia analitikoren arunakvum  D e ⊂G e : Dicuq ρ(∂D, ∂G) = r : Kareli  enadrel r > 0 , orovhet D r = 0 depqum eorem aknhaytoren it : Parz , or D ⊂ Gr ⊂ e r or D -n patkanum  (G) e r bazmuyan or  kapakcva

⊂ (G)

komponentin, orn st naxord lemmayi holomorfuyan tiruy :

§ 19.

Holomorfuyan aan

Bayc ayd depqum na De -n  patkanum nuyn komponentin , e r bazmuyan: Aysteic l bxum , or ρ(∂ D, e ∂ G) e > r: uremn, (G)

H e t a n q 3.5. Ee De -n sahmana ak D tiruyi holo{ ∩

morfuyan aann , apa ∂D ∂ De hatum o datark : A p a c u y c: Kira elov eorem 3.15- D

neri nkatmamb, stanum enq`

e G=D

tiruy{

e 6 ρ(∂ D, e ∂ D) e =0: ρ(∂D, ∂ D)

Ayste havi  a nva , or holomorfuyan tiruyi aan hamnknum  ir het: Aynuhet , D -i sahmana akuyunic he{ e = t um , or ∂D -n kompakt bazmuyun , het abar, ρ(∂D, ∂ D) e = 0 havasaruyunic bxum , or ∂D -n u ∂ D -n hatvum en: 3. Parzaguyn

tiruyneri

holomorfuyan

aan:

Aym ancnenq oro parzaguyn tiruyneri holomorfuyan aani nkaragrman:

 e o r e m 3.16. Ee D -n

eynharti lriv tiruy , apa nra logarimoren u ucik aan holomorfuyan a{ ann : bL D

A p a c u y c: Amen mi funkcia, or holomorf  D -um,

analitikoren arunakvum  Db L -i mej: Aydpisi arunakum katarvum  f -i eylori arqi mijocov, ori zugamituyan tiruy, inpes haytni  eorem 1.2-ic, logarimoren u u{ cik : Mnum  apacucel, or Db L - holomorfuyan tiruy : st eorem 2.6-i ditouyan, dra hamar bavakan  nel funkcianeri or  F ⊂ O(D) ntaniq, ori nkatmamb Db L - F -u ucik : Orpes aydpisi ntaniq karo  a ayel cz k = = cz1k1 · · · znkn miandamneri bazmuyun (tes xndir 2.3):

Glux ԼԼԼ. Ps dou ucik tiruyner

 e o r e m 3.17. TB xoovaka tiruyi H(TB ) holomor{ fuyan aan hamnknum  nra O(TB ) u ucik aani het, aysinqn` H(TB ) = O(TB ) = TO(B) :

A p a c u y c: Katarenq

lracuci enadruyun, or miaer : st § 15.5 -i O(TB ) = TO(B) tiruy ps { dou ucik  , uremn, holomorfuyan tiruy : Qani or H(TB ) -n handisanum  TB -n parunako amena oqr holomorfuyan tiruy, apa H(TB ) ⊂ O(TB ) : Dicuq TB1 - H(TB ) -i mej parunakvo amename xoovaka tiruyn : Aydpisi tiruy goyuyun uni norhiv enadruyan, or H(TB ) -n miaer : Apacucenq, or TB1 = H(TB ) : Katarenq hakaso enadruyun. dicuq TB1 ̸= H(TB ) : Ayd depqum ∂TB1 -i vra kgtnvi z 0 ket, or H(TB ) -i hamar nerqin ket  ori hamar ∆H(T( ) (z 0 ) > 0 : Kartan{Tuleni eoremic het um , or H(TB ) -n xoovaka tiruy : Owremn` H(TB ) -n

∆H(T( ) (x0 +iy) = ∆H(T( ) (z 0 ) > 0,

kamayakan

|y| < ∞

depqum:

Aysteic het um , or TB1 - kareli  me acnel aynpes, or me{

acva tiruy lini xoovaka parunakvi H(TB ) -i mej: Bayc da hakasum  TB1 -i amename linelun: Owremn TB1 = H(TB ) TB1 - ps dou ucik : Inpes haytni , xoovaka tiruy ps dou ucik  ayn miayn ayn depqum, erb nra himq u ucik , aysinqn B1 - u ucik : Aysteic, havi a nelov, or TB ⊂ TB1 = H(TB ) ⊂ O(TB ) = TO(B) ,

ezrakacnum enq, or

B1 = O(B) :

S a h m a n u m 3.4. Ee

{ } G = z : |zn | < R(e z ), ze ∈ B

(3.33)

§ 19.

Holomorfuyan aan

Hartogsi tiruy , apa

G -i

harmonik aan kanvanenq

logarimoren plyurisuper{

{ } G∗ = z : |zn | < eV (ez ) , ze ∈ B

(3.34)

tesqi Hartogsi tiruy, orte V (ez ) - ln R(ez ) -i amena oqr plyurisuperharmonik maorantn  B tiruyum:

 e o r e m 3.18. Ee G -n (3.33) Hartogsi tiruyn , apa G -i holomorfuyan aan hamnknum  nra (3.34) logarimoren plyurisuperharmonik aani het: A p a c u y c: Iroq, ee f - holomorf  G -um, apa na ver{

lu vum  Hartogsi arqi`

f (z) =

n ∑

gk (e z )znk ,

k=0

het abar, na analitikoren arunakvum  ayd arqi { } Gf = z : |zn | < Rf (e z ), ze ∈ B

zugamituyan tiruyi mej: Ayste Rf - Gf tiruyi Hartog{ si a avin  , st lemma 3.1-i ditouyan, ln Rf - plyuri{ superharmonik  B -um ln R(ez ) -i plyurisuperharmonik ma{ orantneric mekn : Owremn, kamayakan f ∈ O(G) funkciayi hamar Ff ⊃ G∗ , qani or G tiruyi H(G) holomorfuyan aan bolor aydpisi Gf -eri hatumn , apa G∗ ⊂ H(G) : Mnum  apacucel, or G∗ -n holomorfuyan tiruy : Da het um  § 17.12 -ic kayi eoremic:

Glux ԼԼԼ. Ps dou ucik tiruyner

Xndirner X n d i r 3.1. Apacucel, or ee f (x) funkcian kisaannd{

hat  ver ic K kompakti vra f (x) < +∞ , apa goyuyun uni nvazo anndhat funkcianeri hajordakanuyun, or gtum  f (x) -in:

X n d i r 3.2. Apacucel, or ee u(z) > 0 funkcian subhar{

monik 

G

tiruyum, apa

up (z) ( p > 1 )

s subharmonik :

X n d i r 3.3. Apacucel, or ee u(z) funkcian subharmonik 

G

tiruyum, apa

eu(z) -

s subharmonik :

X n d i r 3.4. Apacucel, or C 2(G) dasi u(z) funkcian sub{

harmonik 

G

tiruyum ayn miayn ayn depqum, ee ∆u =

∂2u ∂2u ∂2u + ≡ >0: ∂x2 ∂y 2 ∂z ∂ z̄

X n d i r 3.5. Apacucel, or haruyan vra amen mi tiruy

ps dou ucik :

GLUX

1V

INTEGRALAYIN NERKAYACUMNER

§

20. Diferencial 

1. Artaqin  er Cn -um. Enadrvum , or nerco a{

no  irakan Rn tara uyan mej diferencial  eri tesuyan het: Ayd tesuyun, iharke, mnum  ui mej ee Rn - oxari{ nenq Cn = R2n -ov, sakayn kompleqs ka ucva qi a kayuyun a ajacnum  diferencial  eri lracuci hatkuyunner: Orpes irakan koordinatner Cn -um vercnum enq x1 , . . . , xn , y1 , . . . , yn : Cn

tara uyun hamarum enq komnorova aynpes, or dV2n = dx1 ∧ · · · ∧ dxn ∧ dy1 ∧ · · · ∧ dyn ,

(4.1)

orte dV2n - Lebegi a n  R2n -um: Kira elov diferencman perator z = x + iy z̄ = x − iy funkcianeri nkatmamb, stanum enq het yal 1 - er` dzj = dxj + idyj ,

dz̄j = dxj − idyj ,

orteic dxj = (dzj + idz̄j ),

dyj =

Aysteic het um , or amen mi k - α=

∑ I,J

(dzj − idz̄j ) : 2i

Cn -um

AI,J dzI ∧ dz̄J

stanum  (4.2)

Glux Լ7. Integralayin nerkayacumner

tesq: Gumarum katarvum  st ayn I = (i1 , . . . , ip ) J = = (j1 , . . . , jq ) ao p hamapatasxanabar q -indeqsneri, oronc hamar p + q = k , AI,J -er funkcianer en dzI = dzi1 ∧ . . . ∧ dzip ,

dz̄J = dz̄j1 ∧ . . . ∧ dz̄iq :

Trva p q -eri hamar (4.2) gumar kovum  (p, q) erkastiani  , kam parzapes (p, q) - : Amen mi k - miareqoren verlu { vum  (p, k − p) - eri gumari, orte p = 0, . . . , k : Ee f - funkcia , apa nra df =

n ( ∑ ∂f j=1

∂f dzj + dz̄j ∂zj ∂ z̄j

)

diferencial bnakan  ov trohvum  erku maseri` ∂f =

n ∑ ∂f dzj , ∂zj

¯ = ∂f

j=1

Da a ajacnum 

d

n ∑ ∂f dz̄j : ∂ z̄j j=1

peratori verlu uyun` (4.3)

d = ∂ + ∂¯ :

Inpes haytni , (4.2)  i diferencial sahmanvum  het yal kerp` ∑ dAI,J ∧ dzI ∧ dz̄J :

dα =

I,J

Het abar, ¯ dα = ∂α + ∂α,

orte ∂α =

∑ I,J

∂AI,J ∧ dzI ∧ dz̄J ,

¯ = ∂α

∑ I,J

¯ I,J ∧ dzI ∧ dz̄J : ∂A

§ 20.

Diferencial 

¯ - (0, 1) - , usti ∂ ∂¯ Nenq na , or ∂f - (1, 0) - , isk ∂f peratorner amen mi (p, q) - artapatkerum en (p + 1, q) hamapatasxanabar (p, q + 1) erkastiani  eri: Hamaayn (4.3)-i, d2 = 0 diferencial  eri haytni hatku{ yun stanum  het yal tesq` ( ) ¯ + ∂¯2 = 0 : ∂ 2 + ∂ ∂¯ + ∂∂ ( ) ¯ α -n , apa ∂ 2 α -n, ∂ ∂¯ + ∂∂

Ee α -n (p, q) - ∂¯2 α -n unen hamapatasxanabar (p + 2, q) , (p + 1, q + 1) (p, q + 2) erk{ astianner: Owremn, nranc gumar karo  havasarvel zroyi miayn ayn depqum, erb nrancic yuraqanyurn  zro: Het abar` ∂ 2 = 0,

¯ ∂ ∂¯ = −∂∂,

∂¯2 = 0 :

Nermu enq het yal diferencial  er` ω(z) = dz1 ∧ · · · ∧ dzn , ωj (z) = (−1)j−1 dz1 ∧ · · · [j] · · · ∧ dzn ,

orte j = 1, . . . , n , onva

[j] -n

nanakum , or j -rd

ω ′ (z) =

n ∑

dzj

(4.4) andam bac 

zj ωj (z) :

j=1

Ayd nanakumner epet aynqan l hajo en ( ω′ - hiecnum  a ancyal), bayc ndunva en: Ee s = (s1 , . . . , sn ) - irenic nerkayacnum  Cn -um or  ti{ ruyi oork artapatkerum, apa hamapatasxan naxapat{ kerner nanakvelu en ayspes` ω(s) , ωj (s) , ω′ (s) : rinak, ωj (s) = (−1)j−1 ds1 ∧ · · · [j] · · · ∧ dsn , ωj (z̄) - stacvum  (4.4)-ic, ee bolor zi -er oxarinenq z̄i -erov, ω ′ (z̄) =

n ∑ j=1

z̄j ωj (z̄)

Glux Լ7. Integralayin nerkayacumner   uni

(0, n − 1)

erkastian:

P n d u m 4.1. Ayd nanakumnerov hander dz̄j ∧ ωj (z̄) = ω(z̄), dz̄k ∧ ω ′ (z̄) = z̄k ω(z̄), ¯ ′ (z̄) = nω(z̄) : ∂ω

(4.5) (4.6) (4.7)

j = 1, . . . , n, k = 1, . . . , n,

A p a c u y c: (4.5)- het um  sahmanumneric: Qani or dz̄k ∧ ωj (z̄) = 0,

erb

k ̸= j,

apa (4.5)-ic het um  (4.6)-: Baci dranic, qani or ∂¯ [z̄j ωj (z̄)] = dz̄j ∧ ωj (z̄),

apa (4.5)-ic het um  na (4.7)-: 2. Integrum sferayov. Nermu enq het yal nanakumner.

Bn - bac miavor gundn , Sn -` nra ezr3 , orn irenic nerkayac{ num  miavor sfera: Vn -ov nanakvelu  avali a  Cn -um, isk σn -ov` miavor sferayi makeresi a : Sovorabar ays nanakumneri mej n indeqs bac  onvelu,

ee kariq ka hatuk etel tara uyan a oakanuyun:

P n d u m 4.2. Ee p -n

q -n n -indeqsner

en

p ̸= q ,

apa

∫

ζ p ζ̄ q dσ(ζ) = 0 : S

Haaxaki gtagor vum en B2n 5 2n−1 nanakumner, oroncum etvum  gndi sferayi a oakanuyun, mer Bn 5n nanakumneri mej etvum , or nranq Cn -i miavor gundn u sferan en:

§ 20.

Diferencial 

A p a c u y c: Dicuq f ∈ C(B n ) : Fiqsa ζe-i hamar 2π

∫π

e eiθ ζn ) dθ f (ζ,

−π

integral irenic nerkayacnum  f -i mijinacum { } e ζn eiθ ) ∈ S : − π 6 θ 6 π ζ = (ζ,

rjanag i vra: Ev, uremn, nra ov mimyanc havasar en. ∫

∫ f dσ =

S

f -i integralner S

dσ(ζ) 2π

S

∫π

e eiθ ζn ) dθ : f (ζ,

maker uy{ (4.8)

−π

xaxtelov ndhanruyun, kareli  enadrel, or pn ̸= qn : Di{ cuq f (ζ) = ζ p ζ̄ q : Ayd depqum (4.8)-um integral havasar  zroyi:

P n d u m 4.3. Ee p -n n -indeqs , apa ∫

|ζ p |2 dσ(ζ) =

2π n p! , (n − 1 + p)!

|z p |2 dV (z) =

π n p! : (n + |p|)!

∫S B

(4.9) (4.10)

A p a c u y c: Ditarkenq het yal integral. ∫

I=

( ) |z p |2 exp −|z|2 dV2n (z) :

Cn

Ayn havelu hamar nkatenq, or enaintegral artahaytuyun havasar  n ∏

j=1

( ) |zj |2pj exp −|zj |2 ,

Glux Լ7. Integralayin nerkayacumner

, gtvelov Fubinii eoremic, stanum enq I=

n ∫ ∏

( ) |w|2pj exp −|w|2 dV2 (w) :

j=1 C

Yuraqanyur artadri hetuyamb havvum ` ∫

(

) |w|2pj exp −|w|2 dV2 (w) =

C

∫∞ ∫2π

( ) r2pj exp −r2 r dr dφ =

∫∞ tpj exp (−t) dt = πpj !,

=π

orteic stanum enq I = : Inpes haytni , dV2n avali lement b e ayin koordinat{ nerov artahaytvum  het yal bana ov4 . πnp !

dV2n (rζ) = r2n−1 dr dσ(ζ),

(4.11)

Havelov I -n b e ayin koordinatnerov, kstananq ∫∞ n

π p! =

r

2|p|+2n−1 −r2

e

(n − 1 + |p|)! =

∫ |ζ p |2 dσ(ζ) =

dr S

∫

|ζ p |2 dσ(ζ), S

orteic bxum  (4.9)-: Integrelov (4.9)-n s mek angam b e ayin koordinatnerov, kstananq (4.10)-. ∫ |z | dV (z) = B

∫

∫1 p 2

r

2n−1

|(rζ)p |2 dσ(ζ) =

dr S

Ն. Խսմոո. Rօal aոd cոբplօx aոalկտiտ, MօGrոՇ-Hոll, 1987, Cեոքter 8, ex. 6

§ 20.

2π n p! = (n − 1 + p)!

Diferencial 

∫1 r2|p|+2n−1 dr =

P n d u m 4.4. Tei uni het

π n p! : (n + |p|)!

yal haraberakcuyun`

ω(z̄) ∧ ω(z) = (2i)n dV :

Ee f ∈ C(S) , apa ∫

f (ζ) ωj (ζ̄) ∧ ω(ζ) =

(2i)n

f (ζ)ζj dσ(ζ),

(4.13)

f (ζ) dσ(ζ) :

(4.14)

S

∂B

∫

∫

(4.12)

f (ζ) ω ′ (ζ̄) ∧ ω(ζ) =

(2i)

n

∫ S

∂B

A p a c u y c: Qani or dz̄k ∧dzk = 2i dxk ∧dyk , apa havelov ayn tea oxuyunneri qanak, oronq anhraet en dz̄1 ∧ dz1 ∧ . . . ∧ dz̄n ∧ dzn   ω(z̄) ∧ ω(z) -in berelu hamar, kstananq, or ω(z̄) ∧ ω(z)   havasar  (−1)

n(n−1)

(2i)n dx1 ∧ dy1 ∧ · · · ∧ dxn ∧ dyn ,

orn ir herin havasar  (2i)n dx1 ∧ · · · ∧ dxn ∧ dy1 ∧ · · · ∧ dyn = (2i)n dV2n

st (4.1)-i: (4.13)- apacucelu hamar karo enq sahmana kvel C 1 (Cn ) dasi funkcianerov: Ayd depqum φ = f (ζ) ωj (ζ̄) ∧ ω(ζ)

  orova  ∂φ = 0 : Owsti

Cn -um

uni

(n, n − 1)

erkastian: Parz , or

¯ = ∂f ω(ζ̄) ∧ ω(ζ), dφ = ∂φ ∂ ζ̄j

Glux Լ7. Integralayin nerkayacumner

hamaayn Stoqsi bana i u (4.12)-i ∫

∫

φ = (2i)

n B

∂B

Het abar (4.13)- bervum  ∫

∂f dV = ∂ z̄j

B

∂f dV : ∂ z̄j

∫ f (ζ)ζj dσ(ζ)

(4.15)

S

a nuyan: Ee f (z) = z p z̄ p z̄j in-or n -indeqs p -i hamar, apa pndum 4.3-ic het um , or (4.15)- marit : Mnaca bolor miandamneri hamar erku integralnern l havasar en zroyi: Owremn, (4.15)- tei uni bolor bazmandamneri ( z -ic u z̄ -ic) hamar, in apacucum  (4.13)-: Ee (4.13)- kira enq f (ζ) -i oxaren f (ζ)ζ̄j -i nkatmamb gumarenq st j -i 1 -ic min n , apa kstananq (4.14)-: §

21. Koi{Puankarei eorem

Stoqsi bana ov hetuyamb apacucvum  Koi{Puanka{ rei eorem, or Koii eoremi bazmaa ndhanracumn .  e o r e m 4.1 (Koi{Puankare). Dicuq V ⊂ Cn -n sahma{

na ak (n + 1) -a ani maker uy  ktor a ktor oork ∂V ezrov f (z) - holomorf  V -i rjakayum: Ayd depqum ∫

f (z) dz1 ∧ . . . ∧ dzn = 0 : ∂V

¯ =0 A p a c u y c: Qani or f (z) - holomorf , apa ∂f df = ∂f =

n ∑ ∂f dzj : ∂zj j=1

§ 21.

Koi{Puankarei eorem

st Stoqsi bana i` ∫

  n ∑ ∂f  dzj  ∧ dz1 ∧ . . . ∧ dzn = 0 : ∂zj

∫ f (z) dz1 ∧ . . . ∧ dzn =

j=1

V

∂V

marit  na Koi{Puankarei eoremi hakadar pndum, or Morerayi eoremi ndhanracumn : Hiecnenq, or haruyan depqum anndhat funkciayi holomorf linelu hamar bavakan , or nranic integral havasar lini zroyi miayn e ankyun{ neri ezrerov: Nmanapes, tara akan depqum bavakan  sahmana akvel (n + 1) -a ani <prizmanerov>, oronq irencic nerkayacnum en C1zm haruyunum gtnvo ∆m e ankyan [ai , zi ] mnaca C1zi haruyunnerum (i ̸= m) hatva neri Λm dekartyan artadryal:

 e o r e m 4.2.

Ee f (z) funkcian anndhat  D ⊂ Cn tiruyum cankaca

Tm = ∆m × Λm

prizmayi hamar, ori akum patkanum  D -in, ∫ f (z) dz = 0,

(4.16)

∂Tm

apa f - holomorf  D -um: A p a c u y c: Bavakan  apacucel f -i holomorfuyun

kamayakan

a∈D

keti rjakayqum: Fiqsenq a -n ditarkenq ∫z1 F (z) =

∫zn dζ1 . . .

a1

f (ζ) dζn an

Glux Լ7. Integralayin nerkayacumner

funkcian: Na orova anndhat  a -i in-or rjakayqum: Amen mi k -i hamar F - kareli  nerkayacnel ∫zk F (z) =

Fk (ζ) dζk ak

tesqov, orte

Fk (ζ) -ov

nanakva  f -i integral

Λk = [a1 , z1 ] × · · · × [ak−1 , zk−1 ] × [ak+1 , zk+1 ] × · · · × [an , zn ]

bazmuyamb: Fk (ζ) funkcian aknhaytoren anndhat  st ζk -i ak keti in-or Uk rjakayqum st (4.16) paymani kamayakan ∆k b Uk e ankyan hamar ∫ Fk (ζ) dζk = 0 : ∂∆k

Iroq, ayd integral karo  miayn nanov tarbervel ∫

∫ f dζ =

∂∆k ×Λk

f dζ ∂Tk

integralic, orte Tk = ∆k × Λk dζ = dζ1 ∧ . . . ∧ dζn : st Morerayi eoremi mek o oxakani funkcianeri hamar, ays{ teic bxum , or F - holomorf  st zk -i: Ayd datouyunner marit en amen mi k -i hamar, aynpes or F - holomorf  st yuraqanyur o oxakani hamaayn Hartogsi eoremi, na holomorf  a keti rjakayqum: Holomorf  na f -, qani or f (z) =

∂ n F (z) : ∂z1 · · · ∂zn

§ 22.

§

Martineli{Boxneri bana 

22. Martineli{Boxneri bana 

 e o r e m 4.3. Dicuq D ⊂ Cn sahmana ak tiruy  ktor

a ktor oork ezrov, f (z) - holomorf  D -um anndhat  D -um: Tei uni integralayin nerkayacum` (n − 1)! f (z) = (2πi)n

∫

f (ζ) ΩM B (z, ζ),

z ∈ D,

(4.17)

∂D

orte ΩM B (z, ζ) =

ω ′ (ζ̄ − z̄) ∧ ω(ζ) |ζ − z|2n

kovum  Martineli{Boxneri koriz: A p a c u y c: Nax stugenq, or ΩM B (z, ζ) Martineli{Box{

neri koriz ak 

Cn \ {z} -um:

Iroq`

} n ∑ ζ̄k − z̄k dΩM B (z, ζ) = d 2n ωk (ζ̄) ∧ ω(ζ) = |ζ − z| k=1 ( ) n ∑ ∂ ζ̄k − z̄k = ω(ζ̄) ∧ ω(ζ) = ∂ ζ̄k |ζ − z|2n k=1 ) n ( ∑ n(ζ̄k − z̄k )(ζk − zk ) = − ω(ζ̄) ∧ ω(ζ) = 0 : |ζ − z|2n |ζ − z|2n+2 k=1 {

Aynuhet ` d [f (ζ)ΩM B (z, ζ)] = df (ζ) ∧ ΩM B (z, ζ) = ∂f (ζ) ∧ ΩM B (z, ζ) = 0 :

Verjin havasaruyun het um  nranic, or ΩM B (z, ζ) dife{ rencial   parunakum  bolor dζk -er: Kira elov Stoqsi ba{ na  D \ B(z, ε) -um, stanum enq ∫

∫

f (ζ) ΩM B (z, ζ) = ∂[D\B(z,ε)]

d [f (ζ) ΩM B (z, ζ)] = 0, D\B(z,ε)

Glux Լ7. Integralayin nerkayacumner kam

∫

∫ f (ζ) ΩM B (z, ζ) −

∂D

Integral

f (ζ) ΩM B (z, ζ) = 0 : ∂B(z,ε)

∂B(z, ε) -ov ε−2n

havasar  ∫

f (ζ)ω ′ (ζ̄ − z̄) ∧ ω(ζ),

∂B(z,ε)

or, inpes het um  (4.14)-ic, teaaric u gumic heto ha{ vasar  ∫

(2i)n f (z + εζ)ω (ζ̄) ∧ ω(ζ) = ′

∫ f (z + εζ) dσ(ζ) : S

∂B

norhiv f -i anndhatuyan havi a nelov, or miavor 2π n n sferayi aval C -um havasar  , aj mas aknhay{ toren gtum  bana : Erb

n = 1,

(2πi)n f (z) -i, (n − 1)!

(n − 1)!

erb

ε → 0,

stanum enq (4.17)

koriz vera vum  Koii korizi` ΩM B (z, ζ) =

dζ , ζ −z

, uremn, (4.17)- vera vum  Koii bava i: Koii bava  uni erku kar or hatkuyun` 1. na universal  ayn imastov, or marit  bavakanaa

oork ezr uneco bolor tiruyneri hamar, nd orum Koii korizi tesq kaxva  tiruyic, 2. ayd koriz holomorf  st z -i:

§ 22.

Martineli{Boxneri bana 

Martineli{Boxneri bana  nva hatkuyunneric uni miayn a ajin: at o oxakani holomorf funkcianeri hamar i ha{ jovel stanal ayd erku hatkuyunnerov tva integralayin bana : Nranq kam universal en, bayc o holomorf korizov, kam l unen holomorf koriz, bayc universal en: Koii bana  ndhanracvum  anndhat diferenceli funk{ cianeri hamar: Da Pompeyui (Koi{Grini) bana n ` ∫

f (z) = 2πi

∂D

f (ζ) dζ − ζ −z 2πi

∫ D

∂f dζ̄ ∧ dζ : ∂ ζ̄ ζ − z

(4.18)

A ajanum  bnakan xndir. stanal (4.17)-i ndhanracum an{ ndhat diferenceli funkcianeri hamar aynpes, or n = 1 dep{ qum na hamnkni (4.18)-i het: Goyuyun uni Martineli{Box{ neri bana oork funkcianeri hamar`

 e o r e m 4.4.

Dicuq D ⊂ Cn sahmana ak tiruy  ktor a ktor oork ezrov f ∈ C 1 (D) : Ayd depqum bolor z ∈ D -eri hamar (2πi)n f (z) = (n − 1)!

∫

∫

¯ (ζ) ∧ ΩM B (z, ζ) : ∂f

f (ζ) ΩM B (z, ζ) −

(4.19)

D

∂D

A p a c u y c: I tarberuyun eorem 4.3-um ditarkva

depqi,

f (ζ) ΩM B (z, ζ)

  arden ak .

¯ (ζ) ∧ ΩM B (z, ζ), df (ζ) ∧ ΩM B (z, ζ) = ∂f

∫

Stoqsi bana ic stanum enq` ∫

f (ζ) ΩM B (z, ζ) − ∂D

∫

f (ζ) ΩM B (z, ζ) =

∂B(z,ε)

¯ (ζ) ∧ ΩM B (z, ζ) : ∂f

D\B(z,ε)

(4.20)

Glux Լ7. Integralayin nerkayacumner

Integral

∂B(z, ε) -ov, inpes eorem 4.3-um, gtum  (2πi)n f (z) -in erb ε → 0 , isk integral D \ B(z, ε) -ov gtum (n − 1)!  integralin amboj D -ov, orovhet ΩM B koriz patkanum  L1 (D) -in: Ayspisov, ancnelov sahmani, (4.20)-ic stanum enq

(4.19)-:

Isk aym grenq Martineli{Boxneri bana  mi ayl tesqov` f (z) =

(n − 1)! 2π n

∫

1 − ⟨ζ, z⟩ f (ζ) dσ(ζ)− |ζ − z|2n

S

−

(n − 1)! πn

∫ B

orte

⟨f (w), w − z⟩ =

n ∑ j=1

⟨f (w), w − z⟩ dV (w), |w − z|2n

fj (w)(w̄j − z̄j ) :

(4.21)

(4.21)- stacvum 

(4.19)-ic, ee havi a nenq, or ω ′ (ζ̄ − z̄) =

n ∑

(ζ̄j − z̄j )ωj (ζ̄) = ω ′ (ζ̄) −

j=1

n ∑

z̄j ωj (ζ̄),

j=1

gtvenq (4.12) { (4.14) a nuyunneric:

§

23. Lerei bana 

P n d u m 4.5. Dicuq s -n u t -n artapatkerum en Ω ∈ Cn

bac bazmuyun Cn \{0} -i mej oork en: Ee goyuyun uni g : Ω 7→ C aynpisin, or t = gs , apa ω ′ (t) = g n ω ′ (s) :

(4.22)

§ 23.

Lerei bana 

A p a c u y c: Nax hamozvenq, or {z ∈ Cn : z1 ̸= 0} bazmu{

yan vra

z1−n ω ′ (z)

( =d

z2 z1

)

( ∧ ... ∧ d

Aj masum gtnvo   handisanum  kargavorva

(

d

zk z1

)

zn z1

n−1

= z1−1 dzk − z1−2 zk dz1 ,

diferencial  eri artadryal: Qani or (4.23)-i aj kom het yal  eri

) :

hat bnakan  ov

k = 2, . . . , n dz1 ∧ dz1 = 0 ,

z1−n+1 dz2 ∧ . . . ∧ dzn = z1−n z1 ω1 (z)

gumarn , avelacva s

n−1

(4.23)

apa (4.24)

andamner`

− z1−n zk dz2 ∧ . . . ∧ dzk−1 ∧ dz1 ∧ dzk+1 ∧ . . . ∧ dzn = = (−1)k−1 z1−n zk dz1 ∧ . . . [k] . . . ∧ dzn = z1−n zk ωk (z),

(4.25)

orte k = 2, . . . , n : Irar het gumarelov (4.24)-n u (4.25)-, stanum enq (4.23)-: Parz , or g -n Ω -um zroner uni: Ω -i ayn enabazmuyan vra, orte s1 ̸= 0 , bavararvum  t1 ̸= 0 payman s: Owremn` ) ( ) s2 sn ω (s) = ∧ ... ∧ d , s1 s1 ( ) ( ) t2 tn ω ′ (t) = tn1 d ∧ ... ∧ d : t1 t1 ′

(

sn1 d

(4.26) (4.27)

Qani or sk /s1 = tk /t1 t1 = gs1 , apa (4.22)- het um  (4.26)-ic (4.27)-ic erb s1 ̸= 0 : Isk Ω -i ayn keterum, orte s1 = 0 , or  ayl sj baadri  tarber zroyic naxord datouyunner s1 -i

oxaren kira um enq sj -i nkatmamb:

Glux Լ7. Integralayin nerkayacumner

 e o r e m 4.5. Dicuq D -n sahmana ak tiruy  Cn -um

ktor a ktor oork ezrov, z ∈ D fiqsa  C 1 -artapatkerum , aynpisin, or

φ : ∂D 7→ Cn

⟨ζ − z, φ(ζ)⟩ ̸= 0,

bolor ζ ∈ ∂D hamar: Ayd depqum ∫

(n − 1)! f (z) = (2πi)n

f (ζ) ∂D

∩

ω ′ (φ̄(ζ)) ∧ ω(ζ) n ⟨ζ − z, φ(ζ)⟩

(4.28)

amen mi f ∈ O(D) C(D) funkciayi hamar: A p a c u y c: Kira elov (4.22) a nuyun t=

ζ̄ − z̄ |ζ − z|2

s = ζ̄ − z̄

zuygi nkatmamb, kstananq ω′

(

ζ̄ − z̄ |ζ − z|2

) =

ω ′ (ζ̄ − z̄) : |ζ − z|2n

Owremn, (4.17)- kareli  grel (n − 1)! f (z) = (2πi)n

∫

f (ζ) ω ∂D

tesqov: Isk aym ker uy`

C2n

′

(

ζ̄ − z̄ |ζ − z|2

) ∧ ω(ζ)

tara uyan mej ditarkenq het yal ak ma{

{ } ζ̄ − z̄ 2n γ0 = (ζ, η) ∈ C : ζ ∈ ∂D, η = , |ζ − z|2

orn anvanum en Martineli{Boxneri cikl: Martineli{Boxneri bana  grvum  het yal tesqov` (n − 1)! f (z) = (2πi)n

∫

γ0

f (ζ) ω ′ (η) ∧ ω(ζ) :

(4.29)

§ 23.

Lerei bana 

Nman  ov nermu elov { } γ = (ζ, η) ∈ C2n : ζ ∈ ∂D, η = φ(ζ)

Lerei cikl, khamozvenq, or (4.28)-i aj mas grvum  (n − 1)! (2πi)n

∫

f (ζ) ω ′ (η) ∧ ω(ζ)

(4.30)

γ

tesqov: Mnum  apacucel, or (4.29)-um (4.30)-um integralnern irar havasar en: Dra hamar nax nkatenq, or γ0 γ cikler gtnvum en { } T = (ζ, η) ∈ C2n : ⟨ζ − z, η⟩ = 1

maker uyi vra: Cuyc tanq, or T -i vra nranq irar homologik en, aysinqn, nranc vra kareli

el <aan>: Da kareli  ka{ tarel, miacnelov γ0 -n γ -n irar het uagi hatva nerov, ayn ` { } ζ̄ − z̄ Q = (ζ, η) : ζ ∈ ∂D, η = tφ(ζ) + (1 − t) , 06t61 : |ζ − z|2

Hamozvenq, or f (ζ) ω′ (η) ∧ ω(ζ)   T -i vra ak : Iroq, havi a nelov or f - holomorf , stanum enq, or { } d f (ζ)ω ′ (η) ∧ ω(ζ) = ∂f (ζ) ∧ ω ′ (η) ∧ ω(ζ) + f (ζ) · ∂ω ′ (η) ∧ ω(ζ)

 n uni (2n, 0) erkastian: Isk qani or T -i kompleqs a o{ akanuyun havasar  2n − 1 , apa nra vra 2n holomorf diferencialner g oren kaxyal en nranc artaqin artadryal havasar  zroyi: Owremn d {f (ζ)ω′ (η) ∧ ω(ζ)} = 0 Stoqsi ba{ na i norhiv ∫

′

∫

f (ζ) ω (η) ∧ ω(ζ) = ∂T

T

{ } d f (ζ)ω ′ (η) ∧ ω(ζ) = 0 :

Glux Լ7. Integralayin nerkayacumner Myus komic, havi a nelov, or ∫

f (ζ) ω ′ (η) ∧ ω(ζ) −

γ0

∫

∂T = γ0 − γ ,

stanum enq

f (ζ) ω ′ (η) ∧ ω(ζ) = 0 :

(4.31)

γ

(4.29), (4.30) (4.31)-ic het um ` (n − 1)! f (z) = (2πi)n

∫

f (ζ) ω ′ (η) ∧ ω(ζ),

γ

or, inpes tesanq, hamareq  (4.28)-in: Isk aym ditarkenq sahmana ak g oren u ucik D tiruy -ezrov ρ oroi funkciayov: Da nanakum , or ρ ∈ C 2 (Cn ) D -n ayn bazmuyunn , orte ρ < 0 , isk ρ -i gradient zro i da num, aysinqn C2

(

N (ζ) =

∂f (ζ) ∂f (ζ) ∂ ζ̄1 ∂ ζ̄n

)

vektor bavararum  N (ζ) ̸= 0 paymanin bolor ζ ∈ ∂D -eri hamar: Ee ζ ∈ ∂D , apa D -i g oren u ucikuyun nanakum , or (n − 1) -a ani kompleqs hiperharuyun, or oa um  ∂D -n ζ ketum, i hatvum D -i het: Owremn` ⟨ζ − z, N (ζ)⟩ ̸= 0,

z ∈ D, ζ ∈ ∂D :

Ayspisov, menq ezrakacnum enq, or N (ζ) -n karo  katarel φ(ζ) -i der eorem 4.5-um menq stanum enq het yal eorem.

 e o r e m 4.6. Dicuq D -n sahmana ak g oren u ucik tiruy  C 2 -ezrov, ρ oroi funkciayov (

N (ζ) =

∂f (ζ) ∂f (ζ) ∂ ζ̄1 ∂ ζ̄n

)

§ 24.

Veyli bana  ∩

gradientov: Ayd depqum cankaca f ∈ O(D) C(D) funk{ ciayi hamar f (z) =

(n − 1)! (2πi)n

∫ ∂D

ω ′ (N̄ (ζ)) ∧ ω(ζ) , ⟨ζ − z, N (ζ)⟩ n

f (ζ)

z∈D:

(4.32)

Masnavor depqum, erb D = B miavor gundn , karo enq vercnel ρ(z) = |z|2 − 1 : Ayd depqum N (ζ) = ζ (4.32)-ic stanum enq Koii integralayin bana  gndi hamar` (n − 1)! f (z) = (2πi)n

∫ f (ζ) ∂D

ω ′ (ζ̄) ∧ ω(ζ) : (1 − ⟨z, ζ⟩)n

Havi a nelov (4.14)-, ays bana  kareli  grel mi ayl tesqov` (n − 1)! f (z) = 2π n

∫ S

§

f (ζ) dσ(ζ) : (1 − ⟨z, ζ⟩)n

24. Veyli bana 

Hetagayi hamar mez petq  galu Heferi eorem`

 e o r e m 4.7 (Hefer). Dicuq D -n holomorfuyan tiruy 

Cn -um

unen bolor

χ ∈ O(D) : Ayd depqum D × D tiruyum goyuyun holomorf q1 (ζ, z), . . . , qn (ζ, z) aynpisi funkcianer, or ζ, z ∈ D keteri hamar tei uni n ∑ χ(ζ) − χ(z) = qj (ζ, z)(ζj − zj ) j=1

verlu uyun:

Glux Լ7. Integralayin nerkayacumner

ndhanur depqum Heferi eoremi apacuyc het  menq na kndunenq a anc apacuyci: Masnavor depqerum, erb χ - bazmandam  kam l D -n eynharti tiruy , ayd eorem ha{ marya aknhayt :

S a h m a n u m 4.1. Dicuq

tiruyum orova en χ1 , . . . , χN , N > n holomorf funkcianer: ∆ tiruy kovum  analitik polidr, ee ∆ b D D ⊂ Cn

∆ = {z ∈ D : |χi (z)| < 1,

i = 1, . . . , N } :

(4.33)

Analitik polidr kovum  Veyli polidr, ee 1) nra bolor nister

{ } σi = z ∈ G : |χi (ζ)| = 1

(2n − 1) -a ani

bazma uyunner en,

2) cankaca k , 2 6 k 6 n tarber <nisteri> hatumneri a{

oakanuyun (2n − k) -ic o avel : n -a ani σi1 ...in = {z : z ∈ D,

|χis (z)| = 1,

<koeri> miacum kovum  polidri henq σ=

∪

s = 1, . . . , n} nanakvum  σ -ov`

σi1 ...in :

i1 <···<in

Ayd <koer> hamarum enq bnakan  ov komnorova , aysinqn, hamapatasxan σi1 , . . . , σin <nisteri> hajordelu kargov oro{ va : st Heferi eorem 4.7-i χi (ζ) − χi (z) =

n ∑

qij (ζ, z)(ζj − zj )

i = 1, . . . , N :

j=1

Nanakenq Qi1 ...in -ov:

(qij ) , i = i1 , . . . , in , j = 1, . . . , n ,

matrici oroi

§ 24.

Veyli bana 

 e ∩o r e m 4.8 (Veyl).

∈ O(∆)

yacvum 

C(∆) :

Dicuq ∆ -n Veyli tiruy  f ∈ Cankaca z ∈ ∆ ketum f funkcian nerka{

f (z) = (2πi)n

∫

∑

f (ζ)Qi1 ...in (ζ, z) dζ n ∏ [χik (ζ) − χik (z)]

i1 <···<in σ i1 ...in

(4.34)

k=1

integralayin bana ov: A p a c u y c: Lerei (4.28) bana um hamapatasxan  ov

ntrenq q artapatkerum: Polidri ∂∆ ezr bakaca  N N ∪ nisteric` ∂∆ = σi : Yuraqanyur σi nisti hamar vercnenq i=1





( )   φi = φi1 , . . . , φiN =  n ∑

qi1

qij (ζj − zj )

j=1

n ∑

qin qij (ζj − zj )

  : 

j=1

Dicuq γi = φi (σi ) ( φi : σi 7→ γi ): Parzuyan hamar apacuyci arunakuyun kkatarenq n = 2 depqi hamar: γi -eri miacum cikl , orovhet σi1 ,i2 = ∩ = σi1 σi2 koeri vra orova : Oroenq ayn het yal kerp: Nanakenq` { } γi1 ,i2 = (ζ, η) ∈ C4 : ζ ∈ σi1 ,i2 η = tφi1 + (1 − t)φi2 , 0 6 t 6 1 : γi1 ,i2 3 -a ani

maker uyner irar het <sosnum en> γi1 tarber ktorner ardyunqum stacvum  ak maker uy` cikl, or knanakenq γ -yov. γi 2

γ=

(N ∪ i=1

) γi

∪

(

∪ i1 <i2

) γi1 ,i2

:

Glux Լ7. Integralayin nerkayacumner

Dvar  stugel, or γ -n gtnvum  { } (ζ, η) ∈ C2n : ⟨ζ − z, η̄⟩

maker uyi vra: it aynpes, inpes da arvel  Lerei eoremi apacuyci amanak, cuyc  trvum, or γ -n homologayin  Marti{ neli{Boxneri ciklin, uremn f (z) = (2πi)2

∫

f (ζ) ω ′ (η) ∧ ω(ζ) :

(4.35)

γ

Aym nkatenq, or (4.35)-um integralnern st γi -i cikli ktorneri havasar en zroyi, orovhet φi -n holomorf  st ζ -i: Havenq ayd integralnern st mnaca γi1 ,i2 ktorneri` ∫

(2πi)2

f (ζ) (η1 dη2 − η2 dη1 ) ∧ dζ = γi1 ,i2

= (2πi)2 −

(

∫1 dt

+ (1 −

= (2πi)2

)(

φi11

σi1 ,i2

+

−

φi12

)(

) φi21 − φi22 −

)] dζ =

]

tφi12 φi21

∫ f (ζ) σi1 ,i2

tφi11 + (1 − t)φi12

[ f (ζ) (1 − t)φi12 φi21 − tφi11 φi22 −

dt

t)φi22 φi11

= (2πi)2

t)φi22 ∫

∫1

− (1 −

f (ζ)

σi1 ,i2

tφi21

[(

∫

∏

dζ = Qi1 ,i2 dζ

:

(χik (ζ) − χik (z))

k=1

Aysteic (4.35)-ic stanum enq (4.34)-

n=2

depqi hamar:

§ 24.

Veyli bana 

Berenq mi ayl apacuyc: Elnenq Martineli{Boxneri banae{ vic, or ditarkvo depqum uni het yal tesq` f (z) =

n ∫ 1 ∑ (ζ̄1 − z̄1 )dζ̄2 − (ζ̄2 − z̄2 )dζ̄1 f (ζ) ∧ dζ : (2πi) |ζ − z|4 i=1 σ

(4.36)

i

Nax nkatenq, or (4.36)-um enaintegral artahaytuyun { grit  , na irenic nerkayacnum  Wi (ζ, z) =

|ζ −

z|2 [W

P1i P2i dζ i (ζ) − Wi (z)] ζ̄1 − z̄1 ζ̄2 − z̄2

(4.37)

 eric yuraqanyuri diferencial: Katarenq hamapatasxan havumner, parzuyan hamar enadrelov z = 0 nanakelov Wi = Wi (ζ) − Wi (0) , kstananq {

) ζ1 dζ̄1 + ζ2 dζ̄2 ( i P1 ζ̄2 − P2i ζ̄1 + |ζ| } ) 1 ( i i + 2 P1 dζ̄2 − P2 dζ̄1 ∧ dζ = |ζ| ) )( i 1 { ( i + P ζ̄ − P ζ̄ = − ζ d ζ̄ + ζ d ζ̄ Wi |ζ|4 )} )( ( ∧ dζ = + ζ1 ζ̄1 + ζ2 ζ̄2 P1i dζ̄2 − P2i dζ̄1

dΩi = Wi

−

)( ) 1 ( ζ1 P1i + ζ2 P2i ∧ dζ = 4 ζ̄1 dζ̄2 − ζ̄2 dζ̄1 Wi |ζ| ) 1 ( = 4 ζ̄1 dζ̄2 − ζ̄2 dζ̄1 ∧ dζ, |ζ| =

in pahanjvum r stugel: Nkatenq na , or erb z ∈ ∆ ζ ∈ σi , apa miayn Wi (ζ) − −Wi (z) tarberuyunn , or zro i da num (orovhet |Wi (ζ)| = 1 |Wi (z)| < 1 ), isk mnaca ner in-or keterum zro en da num: Da nanakum , or erb z ∈ ∆ ζ ∈ σi , apa miayn Ωi  n , or

Glux Լ7. Integralayin nerkayacumner

ezakiuyunner uni, isk mnaca Ωj , j ̸= i  er ezaki en, Stoqsi bana  kareli  kira el miayn het yal  ov. ∫

∫

d {f (ζ) Ωi (ζ, z)} =

f (ζ) ΩM B (z, ζ) = σi

N ∫ ∑

f (ζ) Ωi (ζ, z) :

j=1σ ij j̸=i

σi

Ee ayd bolor integralner gumarenq, in pahanjvum  st (4.36) bana i, apa amen mi σij ko handipelu  erku angam, mi angam σi nisti komic, myus angam` σj , nd orum, irar haka ak komnoroumnerov: Owremn N ∫ ∑ i=1 σ

f (ζ) ΩM B (ζ, z) =

N ∫ ∑

f (ζ) {Ωi (ζ, z) − Ωj (ζ, z)} ,

i,j=1σ ij

i

orte gumarum katarvum  st bolor indeqsneri kargavor{ va zuygeri ( i < j ): Mnum  nkatel, or hanelu gor ouyunic (4.37) korizneri mej o analitik maser kratvum en` Ωi − Ωj =

) 1 |ζ1 |2 + |ζ2 |2 ( i j j i P P − P P 1 2 1 2 dζ = |ζ|2 Wi Wj =

P1i Wi Wj P1j

P2i dζ : P2j

Ayspisov, stanum enq f (z) = N ∫ ∑ f (ζ) P1i = j (2πi)2 i,j=1 [Wi (ζ) − Wi (z)] [Wj (ζ) − Wj (z)] P1 σij

in hamnknum  Veyli bana i het

n=2

depqum:

P2i dζ, P2j

§ 25.

§

ungei tiruyner

25. ungei tiruyner

 e o r e m 4.9. Amen mi f

funkcia, or holomorf  (4.33) Veyli tiruyum anndhat  nra akman vra, ayd ti{ ruyum nerkayacvum  ∑

f (z) =

∞ ∑

(4.38)

Ai,s (z) [χi (z)]s

i1 <···<in |s|=0

arqov, or zugamitum  bacarak saraa :

∆ -i

nersum hava{

A p a c u y c: Dicuq K b ∆ , ntrenq aynpisi ρ = ρ(K) < 1

iv, or |χi (z)| < ρ , lu uyun`

i = 1, . . . , N ,

erb

z ∈ K:

Owremn, het yal ver{

∞ ∑

[χi1 (z)]s1 · · · [χin (z)]sn = = n s1 +1 ∏ · · · [χin (ζ)]sn +1 [χik (ζ) − χik (z)] i1 ,...,in =0 [χi1 (ζ)]

k=1

=

∞ ∑ [χi (z)]s

[χi (ζ)]s+I |s|=0

,

kzugamiti bacarak havasaraa , erb (z, ζ) ∈ K × σi1 ,...,in : Teadrelov (4.33) Veyli bana i mej, kstananq (4.38)-, orte orpes Ai,s (z) gor akicner handes en galis` Ai,s (z) = (2πi)n

∫

σi1 ...in

S a h m a n u m 4.2.

f (ζ)Qi1 ...in (ζ, z) [χi (ζ)]s+I

dζ :

tiruy kovum  ungei tiruy, ee amen mi f ∈ O(G) funkcia havasaraa G -i nersum mo{ tarkvum  bazmandamnerov: G

Glux Լ7. Integralayin nerkayacumner

Inpes haytni , mek o oxakani funkcianeri hamar un{ gei eorem pndum . orpeszi G ⊂ C1 tiruy lini ungei tiruy, anhraet  bavarar, or na lini miakap5 : Erb n > 1 , ka ungei tiruyneri aydpisi parz erkraa{

akan bnuagrum. o amen mi miakap tiruy ungei tiruy , o amen mi ungei tiruy miakap :

L e m m a 4.1. Ee G tiruy holomorf u ucik 

bazman{ damneri das xit  O(G) -um, apa G -n bazmandamayin u ucik : A p a c u y c: Dicuq A b G : Ayd depqum AbO(G) b G

amen mi z 0 ∈ G \ AbO(G) keti hamar goyuyun uni f ∈ O(G) aynpisin, or sup |f (z)| < f (z 0 ) : z∈A

Owremn, kgtnvi aynpisi

δ > 0,

or (4.39)

δ + sup |f (z)| < f (z 0 ) : z∈A

Qani or bazmandamneri das xit  aynpisi P bazmandam, or δ |f (z) − P (z)| < ,

O(G) -um,

kareli  vercnel

∪ z ∈ A {z 0 } :

Aysteic (4.39)-ic het um ` P (z 0 ) > f (z 0 ) −

isk da nanakum , or

Tes, r.

Á. Â. Øàáàò,

Ìîñêâà, 1985.

δ δ > + sup |f (z)| > sup |P (z)|, 2 z∈A z∈A

G -n

bazmandamayin u ucik tiruy :

Ââåäåíèå â Êîìïëåêñíûé Àíàëèç

, ÷àñòü 1, Íàóêà,

§ 25.

ungei tiruyner

L e m m a 4.2. Ee G tiruy bazmandamayin u ucik , apa kgtnven aynpisi A b G bazmuyun z (k) keteri hajordakanuyun` z (k) ∈ G,

k = 1, 2, . . . ,

z (k) → z 0 ∈ ∂G,

or cankaca f ∈ O(G) funkciayi hamar tei unen |f (z (k) )| 6 sup |f (z)|,

k = 1, 2, . . .

z∈A

(4.40)

anhavasaruyunner: A p a c u y c: Iskapes, ee G tiruy bazmandamayin

u ucik , apa, inpes het um  sahmanumic, kgtnven aynpisi A b G bazmuyun z (k) ∈ G, k = 1, 2, . . . , keteri hajordaka{ nuyun, or G -um uni sahmanayin keter or amen mi f ∈ O(G) funkciayi hamar bolor z (k) keterum tei unen (4.40) anhava{ saruyunner: Kira elov (4.40) anhavasaruyun koordina{ takan zj , j = 1, . . . , n funkcianeri nkatmamb, hamozvum enq, or z (k) hajordakanuyun sahmana ak : st Bolcano{Va{ yertrasi skzbunqi, ayd hajordakanuyunic kareli  anjatel enahajordakanuyun, or zugamitum  z 0 , aknhaytoren ez{ rayin, ketin:

 e o r e m 4.10.

depqum, erb nra ruy :

e G

G -n

ungei tiruy  ayn miayn ayn holomorfuyan aan ungei ti{

A p a c u y c: Pndman mi kom aknhayt 

bxum  aveli ndhanur astic. ee G -i or  holomorf ndlaynum ungei tiruy , apa G -n s ungei tiruy : Hakadar` dicuq G -n ungei tiruy , apacucenq, or e G -n s ungei tiruy : Dicuq f - holomorf  G -um, ayd

Glux Լ7. Integralayin nerkayacumner

depqum goyuyun uni Pk (z) , k = 1, 2, . . . bazmandamneri ha{ jordakanuyun, or zugamitum  havasaraa G -i nersum f -in: Nanakenq Gf -ov Pk hajordakanuyan havasara{ a zugamituyan amename tiruy: Cuyc tanq, or Gf - holomorf u ucik : Enadrenq haka ak` ayd depqum st lemma 4.2-i goyuyun uni A b Gf , ρ(A, ∂Gf ) = r > 0 z 0 ∈ Gf , ρ(A, ∂Gf ) < r aynpisiq, or kamayakan P bazmandami hamar tei uni P (z 0 ) 6 sup |P (z)| z∈A

anhavasaruyun: st (2.3) anhavasaruyan lemma 2.1-ic` |P (z)| 6 sup |P (z)|, z ∈ U (z 0 , ρ) (4.41) z∈Aρ

bolor ρ -eri hamar, ρ < r : Qani or Aρ b Gf , apa (4.41)-ic het um , or Pk hajordakanuyun havasaraa zugamet  U (z 0 , ρ) -um, het abar, U (z 0 , ρ) ⊂ Gf bolor ρ < r hamar , in hnaravor : Owremn, Gf - bazmandamayin u ucik : Ayspisov, apacucvec, or Gf - holomorfuyan tiruy : Qa{ ni or Gf ⊃ G , apa Ge ⊂ Gf : Owremn, Pk -n havasaraa zu{ gamitum  Ge -um, oroelov aynte mi holomorf funkcia: st miakuyan eoremi, ayd funkcian hamnknum  f -i het: Dranov isk hastatvum , or amen mi funkcia f ∈ O(G) nerkayac{ vum  orpes bazmandamneri hajordakanuyun, or zugami{ tum  havasaraa Ge -i nersum: Da nanakum , or Ge -n un{ gei tiruy :

 e o r e m 4.11 (Veyl). Orpeszi G -n lini e G

ungei tiruy, holomorfuyan aan{

anhraet  u bavarar, or nra  lini bazmandamayin u ucik: A n h r a  e  t u  y u n: Dicuq G -n ungei tiruy :

st naxord eoremi Ge -n s ungei tiruy  mnum  kira el lemma 4.1:

§ 26. ∂¯ -xndir

B a v a r a r u  y u n: Dicuq Ge -n bazmandamayin

u ucik tiruy  K b G : Owremn KP b G , vercnenq Veyli bazmandamayin polidr D aynpisin, or Kb P b D b G : Amen mi ζ ∈ ∂D keti hamar goyuyun uni Pζ bazmandam, ori hamar |Pζ (ζ)| > 1 > max |Pζ | : ճP K

st anndhatuyan ays anhavasaruyun pahpanvum  ζ -i in-or Vζ rjakayqum` |Pζ (z)| > 1 > max |Pζ |, ճP K

z ∈ Vζ :

Vζ rjakayqer a kum en ∂D kompakt bazmuyun: ntrenq verjavor vov Vζ1 , . . . , VζN rjakayqer, oronq s a kum en ∂D -n, o Pi -er ( 1 6 i 6 N ) linen hamapatasxan bazmandamner: Ka ucenq ∆′ = {z : |Pi (z)| < 1,

i = 1, . . . , N }

bac bazmuyun vercnenq nra ∆ kapakcva komponent, or parunakum  K -n: Parz , or ∆ -n Veyli bazmandamayin poli{ dr : Kira elov eorem 4.9-, stanum enq (4.38) verlu uyun f -i hamar, or K -i vra zugamitum  havasaraa : Qani or (4.38) arqi andamner mer depqum bazmandamner en, stac{ vum , or K -i vra f - havasaraa motarkvum  bazman{ damnerov: Ayspisov, Ge -n, isk nra het na G -n, ungei ti{ ruy :

§

26. ∂¯-xndir

1. andak teekuyunner mi o oxakani funkcianeri tesuyunic. Mer npatakneri hamar anhraet  mek o o{

Glux Լ7. Integralayin nerkayacumner

xakani funkcianeri tesuyunic erku eorem, oronq kberenq apacuycnerov hander:

 e o r e m 4.12 (Koi{Grini bana ). Dicuq G -n haru{

yan vra verjavor vov oork korerov sahmana akva

tiruy : Ee u ∈ C 1 (G) , apa u(z) = 2πi

∫

∂G

u(ζ) dζ + ζ −z 2πi

∫

G

∂u dζ̄ ∧ dζ , ∂ ζ̄ ζ − z

z∈G:

(4.42)

A p a c u y c: Fiqsa z keti hamar ditarkenq Uε = {ζ : |ζ − z| 6 ε} :

ntrenq ε - aynqan oqr, or tiruyum kira enq ∫

Uε ⊂ G : ∫

Gε

∂G

G ε = G \ Uε

∫ φ−

dφ =

Aynuhet φ

∂Uε

u(ζ) dζ

 i nkatmamb: Qani or Stoqsi bana  φ = ζ −z ζ −z funkcian holomorf  Gε -um φ -n parunakum  dζ diferencial, apa het abar`

¯ = ∂u dζ̄ ∧ dζ , dφ = ∂φ ∂ ζ̄ ζ − z

∫ Gε

∂u dζ̄ ∧ dζ = ∂ ζ̄ ζ − z

∫ ∂G

u(ζ) dζ − ζ −z

∫ ∂Uε

u(ζ) dζ : ζ −z

Ancnelov sahmani, erb ε → 0 , aysteic stanum enq (4.42)-, qani or u -i anndhatuyan norhiv integral ∂Uε -ov gtum  u(a) -in, isk integral Gε -ov gtum  integralin amboj G -ov, orovhet (ζ − z)−1 koriz integreli  G -um:

§ 26. ∂¯ -xndir

 e o r e m 4.13. Dicuq G ⊂ C -n bac sahmana ak baz{ muyun , f ∈ C 1 (G) funkcian sahmana ak  u(z) =

2πi

∫

G

Ayd depqum u ∈

f (ζ) dζ ∧ dζ̄, ζ −z

z∈G:

(4.43)

C 1 (G) ∂u(z) = f (z) : ∂ z̄

(4.44)

A p a c u y c: arunakenq f - amboj haruyan vra, ha{

marelov ayn havasar zroyi G -ic durs: Ayd depqum (4.43)- ka{ reli  grel ∫ u(z) =

2πi

C

f (z + ζ) dζ ∧ dζ̄ ζ

tesqov, orteic er um , or u ∈ C 1 (G) , qani or integrali nani tak kareli  a ancel: ∂u Bavakan  = f havasaruyun apacucel fiqsa ka{ ∂ z̄ mayakan a ∈ G keti rjakayqum: Vercnenq ψ ∈ C01 (G) aynpisin, or ψ ≡ 1 a keti or  V rjakayqum: Ayd depqum u = u1 + u2 , orte ∫

ψ(ζ)f (ζ) dζ ∧ dζ̄, ζ −z G ) ∫ ( 1 − ψ(ζ) f (ζ) u2 (z) = dζ ∧ dζ̄ : 2πi ζ −z u1 (z) = 2πi

G

∂u

Qani or 1−ψ(ζ) = 0 V -um, apa 2 = 0 , erb z ∈ V : Katarelov ∂ z̄ ζ 7→ ζ + z o oxakani oxarinum, grenq u1 - u1 (z) = 2πi

∫

ψ(z + ζ)f (z + ζ) G

dζ ∧ dζ̄ ζ

Glux Լ7. Integralayin nerkayacumner

tesqov: Havi a nelov, or ) ) ∂ ( ∂ ( ψ(z + ζ)f (z + ζ) = ψ(z + ζ)f (z + ζ) , ∂ z̄ ∂ ζ̄

verada nalov naxkin o oxakanin, stanum enq ∂u1 (z) = ∂ z̄ 2πi = 2πi

∫ C

∫ C

) dζ ∧ dζ̄ ∂ ( ψ(z + ζ)f (z + ζ) = ζ ∂ ζ̄ ) dζ ∧ dζ̄ ∂ ( ψ(ζ)f (ζ) = ψ(z)f (z) : ζ −z ∂ ζ̄

Verjin havasaruyun het um  (4.42) Koi{Grini bana ic, ee nra mej u -i oxaren vercnenq ψf , isk orpes G tiruy` rjan, orn ir mej parunakum  ψ funkciayi kri: Havi a { nelov, or ψ(z) = 1 u1 (z) = u(z) , aysteic stanum enq (4.44)-: ¯ = f ha{ 2. ∂¯ -xndir bazmaglanum. Nax ditarkenq ∂u

vasarum, orte f - kompakt kriov (0, 1) - , isk u -n` oroneli ¯ = 0 payman anhraet  lu man funkcia : Hiecnenq, or ∂f goyuyan hanar: Ayl ba erov asa , menq uzum enq lu el het yal gerorova havasarumneri hamakarg` ∂u = fj , ∂ z̄j

j = 1, . . . , n,

(4.45)

ori hamar bavararvum en ∂fj ∂fk − = 0, ∂ z̄k ∂ z̄j

hamateeliuyan paymanner:

j, k = 1, . . . , n

(4.46)

§ 26. ∂¯ -xndir

C1

 e o r e m 4.14. Dicuq n > 1 , f - (0, 1) tipi 

 Cn -um dasi gor akicnerov, kompakt K kriov aynpisin, or ¯ =0: ∂f

Aynuhet , dicuq Ω0 -n Cn \ K bac bazmuyan ansahmana{

ak komponentn : Goyuyun uni miak u ∈ C 1 (Cn ) funkcia, or bavararum 

(4.47)

¯ =f ∂u

havasarman u na u(z) = 0 , z ∈ Ω0 paymanin: A p a c u y c: Dicuq f = u(z) = 2πi

∑

fj (z) dz̄j :

∫ f1 (ζ, z2 , . . . , zn ) C

Ka ucenq

dζ ∧ dζ̄ , ζ − z1

z ∈ C,

(4.48)

funkcian: Katarelov o oxakani oxarinum, ayn kareli  nerkayacnel het yal tesqov` u(z) = 2πi

∫ f1 (z1 + ζ, z2 , . . . , zn ) C

dζ ∧ dζ̄ , ζ

z ∈ C,

∂u

orteic het um , or u ∈ C 1 (Cn ) or = f1 st eo{ ∂ z̄1 rem 4.13-i: Erb 2 6 j 6 n , apa, a ancelov integrali nani ∂f ∂f tak gtvelov 1 = j havasaruyunic, kstananq ∂ z̄j

∂u(z) = ∂ z̄j 2πi =

2πi

∂ z̄1

∫ C

∫ C

∂f1 (ζ, z2 , . . . , zn ) dζ ∧ dζ̄ = ∂ z̄j ζ − z1 ∂fj (ζ, z2 , . . . , zn ) dζ ∧ dζ̄ = fj (z) : ζ − z1 ∂ ζ̄1

Glux Լ7. Integralayin nerkayacumner

Verjin havasaruyun het um  Koi{Grini (4.42) bana ic, ee ayn kira enq fj funkciayi nkatmamb fiqsa (z2 , . . . , zn ) -i depqum: Ayspisov, stacvec, or ∂u(z) = fj (z), ∂ z̄j

erb

1 6 j 6 n,

isk da nuynn , in (4.47)-: Masnavorapes, u -n holomorf  Ω0 -um: (4.48)-ic er um , or u(z) = 0 erb |z2 | - bavakanin me  Ω0 -i kapakcva uyunic het um , or u ≡ 0 Ω0 -um: ¯ Ee u1 - or  mi ayl lu um , apa ∂(u−u 1 ) = 0 , uremn, (u− n − u1 ) - amboj funkcia  C -um: Isk ee u1 (z) - s nuynabar zro  Ω0 -um, apa u − u1 ≡ 0 Ω0 -um, orteic het um , or u − u1 ≡ 0 Cn -um: Da l nanakum  lu man miakuyun: D i t o  u  y u n 4.7. Nenq, or n = 1 depqum eorem (4.14)- it  (tes xndir 4.9):

 e o r e m 4.15.

Dicuq G ⊂ Cn bazmaglan Gk har tiruyneri dekartyan artadryal ` G = G1 × · · · × Gn ; fj , j = 1, . . . , n funkcianer patkanum en C 1 (G) -in bava{ rarum en (4.46) paymannerin: Ayd depqum goyuyun uni u ∈ ∈ C 1 (G) funkcia, or bavararum  (4.45) havasarumneri hamakargin: A p a c u y c: Apacuyc katarenq indukciayov st n -i:

Erb n = 1 , pndum arden apacucva  eorem 4.13-um: En{ adrenq, eoremi pndum marit , erb o oxakanneri iv i gerazancum ( n − 1 )-, apacucenq nra marit linel n

o oxakani hamar: Ditarkenq (4.45) hamakargi ∂u = fn ∂ z̄n

§ 26. ∂¯ -xndir

verjin havasarum nanakenq g -ov nra lu um Gn -um, orn irenic nerkayacnum  zn -i funkcia, kaxva ze = (z1 , . . . , zn−1 ) parametric: (4.45) hamakargi lu um oronenq u = g+φ tesqov: Ayd depqum φ -n petq  lini holomorf st zn -i Gn -um, isk mna{ ca koordinatneri nkatmamb Ge = G1 × · · · × Gn−1 tiruyum na petq  bavarari ∂g ∂φ = fk − ≡ hk , ∂ z̄k ∂ z̄k

k = 1, . . . , n − 1

(4.49)

hamakargin: Qani or ∂fj ∂fk = ∂ z̄j ∂ z̄k

apa

∂2g ∂2g , = ∂ z̄j ∂ z̄k ∂ z̄k ∂ z̄j

∂hj ∂hk = , ∂ z̄j ∂ z̄k

j, k = 1, . . . , n,

j, k = 1, . . . , n − 1,

in nanakum , or (4.49) hamakarg bavararum  hamatee{ liuyan paymannerin: st induktiv enadruyan, goyuyun e lu um, or kaxva  zn para{ uni ayd hamakargi φ ∈ C 1 (G) metric: Mnum  hamozvel, or φ -n st zn -i holomorf , isk dra hamar bavakan  stugel, or (4.49)-i aj maser holomorf en st zn -i: Iroq, ∂hk ∂fk ∂2g ∂fk ∂fn = − = − =0 ∂ z̄n ∂ z̄n ∂ z̄n ∂ z̄k ∂ z̄n ∂ z̄k

bolor

k = 1, . . . , n − 1

st (4.46):

Glux Լ7. Integralayin nerkayacumner

§

27. Ke nfunkcia

Dicuq B 2 (Ω) Ω -um ayn holomorf funkcianeri bazmuyunn , oronc hamar  ∥f ∥ = ∥f ∥Ω = 

1/2

∫

|f |2 dV 

Ω

B 2 (Ω) -n L2 (Ω, dV ) tara uyan enatara uyun : Fiqsa z ∈ Ω depqum u 7→ u(z) artapatkerum handi{ sanum  g ayin funkcional B 2 (Ω) -i vra, menq na kanvanenq areq z ketum: Hajord lemman cuyc  talis, or ayd funkcional anndhat  B 2 (Ω) -um:

 e o r e m 4.16. Dicuq a ∈ Ω

U (a, r) b D :

r>0

Ayd depqum

|f (a)| 6

π n/2 rn

iv aynpisin , or

∥f ∥

kamayakan f ∈ B 2 (Ω) funkciayi hamar: A p a c u y c: f (z) funkcian U (a, r) -um verlu enq arqi` f (z) =

∞ ∑

ck (z k − ak ) :

|k|=0

Dicuq

zm − am = ρm eitm : Ownenq` ∫ ∑ ∥f ∥U = ck c̄i (z k − ak )(z̄ i − āi ) dV = U

k,i

§ 27.

=

∑

ck c̄i

=

k

e

i(km −im )tm

∫r ρkmm +im +1 dρm =

dtm

m=1 0

k,i

∑

n ∫2π ∏

Ke nfunkcia

n ∏ r2(km +1) |ck | (2π) , 2(km + 1) m=1

qani or arqi andamner o bacasakan en, apa ∥f ∥2U > |c0 |π n r2n = |f (a)|2 π n r2n :

Het abar` |f (a)| 6

 e o r e m 4.17.

π n/2 rn

∥f ∥U 6

π n/2 rn

∥f ∥ :

B 2 (Ω) -n

handisanum  L2 (Ω, dV ) tara{

uyan ak enatara uyun: A p a c u y c: Dicuq ∥fj − f ∥ → 0 erb j → ∞ , orte fj hajordakanuyun patkanum  B 2 (Ω) -in, isk f ∈ L2 (Ω, dV ) : Petq  apacucel, or f - hamareq  Ω -um or  holomorf funk{ ciayin: Dicuq K -n Ω -i kompakt enabazmuyun : eorem 4.16-ic het um , or goyuyun uni C hastatun, aynpisin, or max |f (z)| 6 C∥f ∥ bolor f ∈ B 2 (Ω) funkcianeri hamar: Het a{ z∈K bar |fj (z) − fk (z)| 6 C∥fj − fk ∥

bolor z ∈ K j, k = 1, 2 . . . : Qani or fj -n fundamental  B 2 (Ω) -um, apa aysteic bxum , or Ω -i kompakt enabazmu{ yunneri vra fj hajordakanuyun havasaraa zugamitum  in-or h funkciayin, or holomorf  Ω -um: Baci dranic, fj hajordakanuyun zugamitum  f -in L2 (Ω, dV ) tara uyan mej: st isi eoremi, goyuyun uni fj -i enahajordakanu{ yun, or ketoren zugamitum  f -in hamarya amenureq Ω -um:

Ayspisov,

Glux Լ7. Integralayin nerkayacumner

hamarya amenureq Ω -um, uremn

f ∈ B 2 (Ω) -in:

f =h

eorem 4.17-ic bxum , or B 2 (Ω) -n hilbertyan tara uyun  ∫

⟨f, g⟩ =

f ḡ dV Ω

nerqin artadryalov: Qani or f 7→ f (z) anndhat artapatke{ rum g ayin funkcional  B 2 (Ω) -um amen mi z ∈ Ω keti hamar, apa hilbertyan tara uyunneri ndhanur tesuyunic hete{ vum , or goyuyun uni miak Kz (ζ) ∈ B 2 (Ω) funkcia aynpisin, or ∫ f (z) =

f (ζ)Kz (ζ) dV (ζ) : Ω

Isk aym -um vercnenq or  uk ronormal bazis: Inpes haytni  (noric hilbertyan tara uyunneri ndhanur tesu{ yunic), B 2 (Ω)

Kz =

∞ ∑

⟨Kz , uk ⟩uk =

k=0

∞ ∑

uk (z)uk ,

k=0

arq zugamitum  B 2 (Ω) -i normov amen mi z ∈ Ω keti hamar: Aysteic, havi a nelov, or <areq ketum> anndhat funk{ cional , stanum enq, or Kz (ζ) =

∞ ∑

uk (z)uk (ζ)

k=0

arq zugamitum  bolor

S a h m a n u m 4.3.

tiruyi ke nfunkcia:

z, ζ ∈ B 2 (Ω)

hamar:

K(z, ζ) = Kz (ζ)

funkcian kovum 

Ke nfunkcian uni het yal hatkuyunner.

Ω

§ 27.

Ke nfunkcia

1. ayn holomorf  st a ajin koordinati st erkrordi,

hakaholomorf 

2. ke nfunkcian hakahamaa , aysinqn`

K(ζ, z) = K(z, ζ) :

Ayspisov, menq apacucecinq ∫

f (z) =

f (ζ)K(z, ζ) dV (ζ)

(4.50)

Ω

integralayin bana  cianeri hamar, orte

B 2 (Ω)

K(z, ζ) =

tara uyan patkano funk{

∞ ∑

uk (z)uk (ζ) :

(4.51)

k=0

Isk aym havenq m i a v o r g n d i ke nfunkcian: Orpes ronormal bazis vercnenq uk (z) = λk z k ,

k = (k1 , . . . , kn ), ki > 0,

miandamneri hamakarg, orte normavoro gor akicnern ntr{ va en aynpes, or ⟨uk , uk ⟩ = 1 : nd orum, uk hamakargi lrivu{ yun het um  ayn banic, or st miandamneri verlu uyun eylori arq , orov, inpes haytni , nerkayacvum  B -um amen mi holomorf funkcia: Aynuhet , havi a nelov (4.10) a nuyu{ n, stanum enq` ∫

⟨uk , uk ⟩ =

z k z̄ k dV = λ2k

λ2k

k! π n = 1, (|k| + n)!

B

orteic λ2k =

(|k| + n)! : k! π n

(4.52)

Glux Լ7. Integralayin nerkayacumner

st (4.51) bana i (4.52)-ic stanum enq KB (z, ζ) =

∞ ∑

λ2k z k ζ̄ k

|k|=0

=

πn

∞ ∑

∞ 1 ∑ (|k| + n)! k k = n z ζ̄ = π k! |k|=0

(m + 1) · · · (m + n)

∑ m! z k ζ̄ k : k!

|k|=m

m=0

Aynuhet , nerqin gumar havasar  ∑ m! ∑ )k )k ( ( m! z k ζ̄ k = z1 ζ̄1 1 · · · zn ζ̄n n = k! k1 ! · · · kn ! |k|=m |k|=m m  n ∑ = zj ζ̄j  = ⟨z, ζ⟩m : j=1

Elnelov het yal tarrakan a nuyunic` ∞ ∑

(m + 1) · · · (m + n)q

m=0

=

havi a nelov, or enq

d = n dq

n! , (1 − q)n+1

q = ⟨z, ζ⟩

KB (z, ζ) =

m

(

1−q

erb

) =

|q| < 1,

ir modulov oqr  mekic, stanum

πn

(

n! 1 − ⟨z, ζ⟩

)n+1 :

Ayspisov, (4.50) bana n ndunum  het yal tesq` f (z) =

n! πn

∫ ( B

f (ζ)dV (ζ) )n+1 : 1 − ⟨z, ζ⟩

Xndirner

Xndirner X n d i r 4.1. Gtnel Cn -um miavor gndi aval: X n d i r 4.2. Gtnel Cn -um miavor sferayi aval: X n d i r 4.3. (B e ayin koordinatnerov integrum Rn -um):

Apacucel, or Rn -um amen mi borelyan f funkciayi hamar tei uni het yal bana ` ∫

∫∞ f dV =

Rn

∫ r

n−1

dr

f (rζ)dσ(ζ) : S

X n d i r 4.4. Apacucel, or Martineli{Boxneri koriz ka{

reli  grel na

(n − 1)! ∑ ∂g dζ̄[k] ∧ dζ (−1)k−1 n (2πi) ∂ ζ̄k n

ΩM B (ζ, z) =

k=1

tesqov, orte g(ζ, z) = (1 − n)−1 |ζ − z|2−2n funkcian Laplasi ∆g = 0 havasarman fundamental lu umn  ζ = z ezakiu{ yamb:

X n d i r 4.5. Cuyc tal, or (4.17) bana  kareli  lracnel

het yal  ov.

∫ f (ζ) ΩM B (z, ζ) = 0,

erb

z∈ /D:

∂D

X n d i r 4.6. Cuyc tal, or ee ezrayin funkcian annd{

hat , apa Martineli{Boxneri integral harmonik funkcia  ∂D -ic durs:

Glux Լ7. Integralayin nerkayacumner

X n d i r 4.7. Gtnel miavor polidiski ke nfunkcian: X n d i r 4.8. Apacucel Heferi eorem 4.7- het yal depqeri

hamar.

1. erb χ -n bazmandam , 2. erb

D -n

eynharti tiruy :

X n d i r 4.9. Berel f ∈ C01(C) funkciayi rinak, ori hamar

goyuyun uni

∂u =f ∂ z̄

havasarman kompakt kri uneco lu um:

Grakanuyun 1.

Á. Â. Øàáàò. Ââåäåíèå â Êîìïëåêñíûé Àíàëèç,

Íàóêà, Ì,

1985. 2.

Â. Ñ. Âëàäèìèðîâ. Ìåòîäû òåîðèè ôóíêöèé ìíîãèõ êîì-

ïëåêñíûõ ïåðåìåííûõ,

Ì., Ôèçìàòãèç, 1964. Ââåäåíèå â òåîðèþ àíàëèòè÷åñêèõ ôóíêöèé ìíîãèõ êîìïëåêñíûõ ïåðåìåííûõ, Ôèçìàòãèç, Ì., 1962.

3.

Á. À. Ôóêñ.

4.

G. M. Henkin, J. Leiterer. Theory of functions on complex

manifolds,

5. 6.

AkademieVerlag, Berlin, 1984.

Ë. Õåðìàíäåð.

Ââåäåíèå â òåîðèþ ôóíêöèé íåñêîëüêèõ

êîìïëåêñíûõ ïåðåìåííûõ,

Ìèð, Ì., 1968.

Ð. Ãàííèíã, Õ. Ðîññè.

Àíàëèòè÷åñêèå ôóíêöèè ìíîãèõ

êîìïëåêñíûõ ïåðåìåííûõ,

Ìèð, Ì., 1969. ôóíêöèé â ïîëèêðóãå, Ìèð, Ì., 1974. n ôóíêöèé â åäèíè÷íîì øàðå èç C , Ìèð,

7. 8.

Ó. Ðóäèí. Òåîðèÿ

9.

Ë. À. Àéçåíáåðã, Á. Ñ. Çèíîâüåâ. Ýëåìåíòàðíûå ñâîé-

Ó. Ðóäèí. Òåîðèÿ

Ì., 1984. ñòâà è èíòåãðàëüíûå ïðåäñòàâëåíèÿ ãîëîìîðôíûõ ôóíêöèé ìíîãèõ êîìïëåêñíûõ ïåðåìåííûõ,

10.

Êðàñíîÿðñê, 1977.

À. Ã. Âèòóøêèí, Çàìå÷àòåëüíûå ôàêòû êîìïëåêñíîãî àíà-

ëèçà.

 ñá.: Ñîâðåìåííûå ïðîáëåìû ìàòåìàòèêè. Ôóíäàìåíòàëüíûå íàïðàâëåíèÿ. Èòîãè íàóêè è òåõíèêè. Ì. 8 (1985), 523.

A arkayakan cank Abeli eorem 16

Analitik arunakuyun 74 Analitik polidr 167 Anhavasaruyun Koii 18, 52 } Yenseni 106 Anorouyan ket 66 Anndhatuyan uyl skzbunq 123 Artapatkerum biholomorf 55, 58 Artapatkerum holomorf 52, 58 Astianayin arqer 14 Avtomorfizm 59 } miavor gndi 59 } polidiski 62 Bazmaglan 11

Bergmani ezr 44 B e ayin bazmuyun 66 Boxneri eorem 78 Diferencial  148

Dirixlei xndir 32, 103 rmityan artadryal 6 aan holomorfuyan 140

} Hartogsi tiruyi 146 } eynharti tiruyi 144 } xoovaka tiruyi 145 eorem argelqi veraberyal 80, 82 } miakuyan 37, 43 Integrum sferayov 151 Lemma miaamanakya arunakman veraberyal 85 Lerei bana  161 L ii   114 Liuvili eorem 44

Bazmandamayin u ucik tiruy 84 Bazmapatik astianayin arq 14 Behenke{ teyni eorem 89 Xist ps dou ucik tiBehenke{Zomeri eorem 91, 134 ruy 130, 131

A arkayakan cank

Xumb avtomorfizmneri 62, 63

} plyurisubharmonik funkciayi hamar 110 Kartan{Tuleni eorem 84, 86 Martineli{Boxneri banaKe nfunkcia 183   158, 161 } gndi 186 Metrika vklidyan 10 } polidiski 188 } polidiskayin 10 Kompleqs haruyun 8 Miakuyan eorem 43 } hiperharuyun 8 } bazmuyun 71 } ui 8 Mittag{Lefleri eorem 67 } a oakanuyun 8 Morerayi eorem 156 Koii bana  38, 166 Multiindeqs 14, 42 } anhavasaruyunner 18, 52 arq astianayin 14 Koi{Grini bana  177, 179 } Lorani 45 Koi{Puankarei eorem 155 Koi{ imani oa o pera- } Hartogsi 47 } Hartogs-Lorani 49 tor 75 } hamase bazmandamnerov 49 Koi-Hadamari eorem 22 ilovi ezr 44 Kuzeni himnaxndir 66 oa o perator 76 Harmonik maorant 103 varci lemman 54, 64 Harnaki eorem 99 Plyuriharmonik funkcia 28, 29  anhavasaruyun 100 Plyurisubharmonik funkHartogsi eorem 24, 157 cia 109 } tiruy 12, 134 } haytani 115 Heferi eorem 166 } mijin areq 110 Henq polidiski 10 } motarkum 112 } polidri 167 Holomorf funkciayi zroner 33 Ps dou ucik tiruy 121 } xoovaka 140 Holomorf u ucikuyun 83 Holomorfuyan tiruy 79, 82 Puankarei eorem 63 Puasoni integralayin banaMaqsimumi skzbunq 43   99

A arkayakan cank Puasoni koriz 99

Funkcia holomorf 23 } he avoruyan 106 eynharti tiruy 11, 144, 188 } kisaanndhat 97 } lriv 11 } meromorf 65 } logarimoren u ucik 16 } plyurisubharmonik 109 ungei tiruy 172 } subharmonik 98 Skalyar artadryal rmityan 9 } u ucik 117 } vklidyan 9 Cn tara uyun 7 Subharmonik funkcia 97 } komnoum 148 } mijin areq 105 F -u ucikuyun 84 } haytani 100 ∂¯ -xndir 176, 179 } bazmaglanum 179 Vayertrasi ps dobazmandam 34 Vayertrasi eorem 42 } naxapatrastakan 34 Veyli bana  166 Veyli polidr 167 Veradrman tiruy 141 Tiruy rjana 12

} xoovaka 13 } g oren u ucik 14 } eynharti 11 } Hartogsi 12 Ow ucik tiruy 136

} funkcia 118 Ow ucikuyun st L ii 91, 92 kayi eorem 135, 146 Fatui rinak 65

PETROSYAN ALBERT ISRAYELI

BAZMA A KOMPLEQS

ANALIZI HIMUNQNER

Buhakan dasagirq

Hrat. xmbagir` M. G. Yavryan Tex. xmbagir` V. Z. Bdoyan

Storagrva  tpagruyan 30. 07. 07 : a s` 60 × 84 1/16: u` fse: Hrat. 10,7 mamul, tpagr. 12,2 mamul = 11,4 paym. mamuli: Tpaqanak` 200: Patver` Er ani hamalsarani hratarakuyun Er an, Al. Manukyan 1 Er ani petakan hamalsarani tpagrakan artadramas, Er an, Al. Manukyan 1