Formula Yensena ye tverdzhennyam u kompleksnomu analizi sho opisuye povedinku golomorfnoyi v kruzi funkciyi v zalezhnosti vid moduliv nuliv ciyeyi funkciyi Tverdzhennya ye vazhlivim zokrema pri vivchenni cilih funkcij Zmist 1 Tverdzhennya 2 Dovedennya 3 Zastosuvannya 4 Uzagalnennya 4 1 Meromorfni funkciyi 4 2 Formula Puassona Yensena 5 LiteraturaTverdzhennya red Nehaj f displaystyle f nbsp ye golomorfnoyu funkciyeyu v oblasti kompleksnoyi ploshini sho mistit zamknutij krug D 0 r displaystyle overline D 0 r nbsp z centrom 0 i radiusom r i a 1 a 2 a N displaystyle alpha 1 alpha 2 ldots alpha N nbsp nuli f displaystyle f nbsp v D 0 r displaystyle overline D 0 r nbsp vrahovuyuchi yih kratnist Yaksho f 0 displaystyle f 0 nbsp ne ye rivnim nulyu to log f 0 k 1 N log r a k 1 2 p 0 2 p log f r e i 8 d 8 displaystyle log f 0 sum k 1 N log left frac r alpha k right frac 1 2 pi int 0 2 pi log f re mathrm i theta mathrm d theta nbsp Ekvivalentno yaksho n r displaystyle n r nbsp poznachaye kilkist nuliv funkciyi f displaystyle f nbsp strogo menshih za modulem r displaystyle r nbsp to log f 0 0 r n s s d s 1 2 p 0 2 p log f r e i 8 d 8 displaystyle log f 0 int 0 r frac n s s mathrm d s frac 1 2 pi int 0 2 pi log f re mathrm i theta mathrm d theta nbsp Dovedennya red Pripustimo spershu sho funkciya f displaystyle f nbsp ne maye nuliv u D 0 r displaystyle overline D 0 r nbsp U comu vipadku vona ne maye nuliv u D 0 r e displaystyle D 0 r varepsilon nbsp dlya deyakogo malogo e displaystyle varepsilon nbsp Oskilki D 0 r e displaystyle D 0 r varepsilon nbsp ye odnozv yaznoyu i f displaystyle f nbsp ne ye rivnoyu nulyu to isnuye funkciya g displaystyle g nbsp sho ye golomorfnoyu v D 0 r e displaystyle D 0 r varepsilon nbsp taka sho f e g displaystyle f e g nbsp Tomu funkciya log f R e g displaystyle log f mathrm Re g nbsp dijsna chastina golomorfnoyi funkciyi ye garmonichnoyu v D 0 r e displaystyle D 0 r varepsilon nbsp Zokrema vona ye garmonichnoyu v D 0 r displaystyle D 0 r nbsp i neperervnoyu v D 0 r displaystyle overline D 0 r nbsp Zgidno vlastivosti serednogo znachennya log f 0 1 2 p 0 2 p log f r e i 8 d 8 displaystyle log f 0 frac 1 2 pi int 0 2 pi log f re mathrm i theta mathrm d theta nbsp Ce zavershuye pershu chastinu dovedennya Pripustimo sho funkciya f displaystyle f nbsp maye nuli v D 0 r displaystyle overline D 0 r nbsp pronumerovani v takij sposib a 1 a m lt r a m 1 a N r displaystyle alpha 1 leq ldots leq alpha m lt r quad alpha m 1 ldots alpha N r nbsp Poznachimog z f z n 1 m r 2 a n z r a n z n m 1 N a n a n z displaystyle g z f z times prod n 1 m frac r 2 overline alpha n z r alpha n z times prod n m 1 N frac alpha n alpha n z nbsp Funkciya g displaystyle g nbsp ye golomorfnoyu v D 0 r e displaystyle D 0 r varepsilon nbsp i ne rivnoyu nulyu v D 0 r displaystyle overline D 0 r nbsp Zgidno pershoyi chastini dovedennya 1 2 p 0 2 p log g r e i 8 d 8 log g 0 log f 0 n 1 N log r a n displaystyle frac 1 2 pi int 0 2 pi log g re mathrm i theta mathrm d theta log g 0 log f 0 sum n 1 N log left left frac r alpha n right right nbsp Tomu dlya zavershennya dovedennya dostatno pokazati sho 0 2 p log g r e i 8 d 8 0 2 p log f r e i 8 d 8 displaystyle int 0 2 pi log g re mathrm i theta mathrm d theta int 0 2 pi log f re mathrm i theta mathrm d theta nbsp Oskilki 0 2 p log g r e i 8 d 8 0 2 p log f r e i 8 d 8 n 1 m 0 2 p log 1 d 8 n m 1 N 0 2 p log 1 e i 8 a n r d 8 displaystyle int 0 2 pi log g re mathrm i theta mathrm d theta int 0 2 pi log f re mathrm i theta mathrm d theta sum n 1 m int 0 2 pi log 1 mathrm d theta sum n m 1 N int 0 2 pi log left left 1 e mathrm i theta overline alpha n r right right mathrm d theta nbsp i poznachivshi a n r e i 8 n displaystyle alpha n re mathrm i theta n nbsp otrimuyemo 0 2 p log 1 e i 8 a n r d 8 8 n 2 p 8 n log 1 e i u d u 0 2 p log 1 e i v d v 0 displaystyle int 0 2 pi log left left 1 e mathrm i theta overline alpha n r right right mathrm d theta int theta n 2 pi theta n log 1 e mathrm i u mathrm d u int 0 2 pi log 1 e mathrm i v mathrm d v 0 nbsp tozh 0 2 p log g r e i 8 d 8 0 2 p log f r e i 8 d 8 displaystyle int 0 2 pi log g re mathrm i theta mathrm d theta int 0 2 pi log f re mathrm i theta mathrm d theta nbsp sho zavershuye dovedennya Zastosuvannya red Fundamentalna teorema algebriFundamentalna teorema algebri stverdzhuye sho kozhen mnogochlen z kompleksnimi koeficiyentami stepenya k displaystyle k nbsp maye k displaystyle k nbsp koreniv vrahovuyuchi kratnist Dlya teoremi isnuye kilka doveden z vikoristannyam idej kompleksnogo analizu Zokrema dlya dovedennya mozhna vikoristati formulu Yensena Nehaj mayemo mnogochlen P X a 0 a 1 X a k X k displaystyle P X a 0 a 1 X dots a k X k nbsp de a k displaystyle a k nbsp ne dorivnyuye nulyu Pripustimo takozh sho a 0 displaystyle a 0 nbsp ne dorivnyuye nulyu Vidobrazhennya z P z displaystyle z mapsto P z nbsp ye ciloyu funkciyeyu tobto golomorfnoyu v C displaystyle mathbb C nbsp Dlya velikih za modulem kompleksnih chisel mayemo P z a k z k displaystyle P z sim a k z k nbsp Zgidno z klasichnimi metodami porivnyannya rozbizhnih integraliv mayemo 1 2 p 0 2 p log P r e i 8 d 8 k log r displaystyle frac 1 2 pi int 0 2 pi log P re mathrm i theta mathrm d theta sim k log r nbsp dd Mnogochlen stepenya k displaystyle k nbsp v C displaystyle mathbb C nbsp maye shonajbilshe k kompleksnih koreniv vrahovuyuchi kratnist Todi kilkist koreniv u kruzi n r displaystyle n r nbsp dlya dostatno velikih r displaystyle r nbsp ye konstantoyu rivnoyu kilkosti koreniv mnogochlena n 0 displaystyle n 0 nbsp Zgidno z formuloyu Yensena1 2 p 0 2 p log P r e i 8 d 8 log P 0 0 r n s s d s n 0 log r Constante displaystyle frac 1 2 pi int 0 2 pi log P re mathrm i theta mathrm d theta log P 0 int 0 r frac n s s mathrm d s n 0 log r text Constante nbsp dd Pislya porivnyannya dvoh ekvivalentnostej n 0 k displaystyle n 0 k nbsp Tobto mnogochlen P displaystyle P nbsp maye k displaystyle k nbsp koreniv vrahovuyuchi kratnist Formula Yensena vikoristovuyetsya dlya ocinennya kilkosti nuliv golomorfnih funkcij A same yaksho f ye golomorfnoyu v kruzi radiusa R z centrom u tochci z0 i yaksho f ye obmezhenoyu chislom M na mezhi kruga todi kilkist nuliv f u kruzi radiusa r lt R z centrom u cij zhe tochci z0 ne perevishuye1 log R r log M f z 0 displaystyle frac 1 log R r log frac M f z 0 nbsp Formula Yensena ye vazhlivoyu u vivchenni rozpodilu znachen cilih i meromorfnih funkcij zokrema teoriyi Nevanlinni Formula Yensena dlya mnogochleniv odniyeyi zminnoyi dozvolyaye obchisliti miru Malera mnogochlena tobto dobutok koreniv mnogochlena z modulem bilshim 1 Uzagalnennya red Meromorfni funkciyi red Formulu Yensena mozhna uzagalniti dlya meromorfnih funkcij u D displaystyle D nbsp Pripustimo sho f z z l g z h z displaystyle f z z l frac g z h z nbsp de g i h ye golomorfnimi u D displaystyle D nbsp z nulyami u tochkah a 1 a n D 0 displaystyle a 1 ldots a n in mathbb D backslash 0 nbsp i b 1 b m D 0 displaystyle b 1 ldots b m in mathbb D backslash 0 nbsp vidpovidno Formula Yensena dlya meromorfnih funkcij maye vid log g 0 h 0 log r m n a 1 a n b 1 b m 1 2 p 0 2 p log f r e i 8 d 8 displaystyle log left frac g 0 h 0 right log left r m n frac a 1 ldots a n b 1 ldots b m right frac 1 2 pi int 0 2 pi log f re i theta d theta nbsp Formula Puassona Yensena red Formula Yensena ye naslidkom bilsh zagalnoyi formuli Puassona Yensena yaka natomist viplivaye z formuli Yensena za dopomogoyu peretvorennya Mebiusa zastosovanogo do z Cyu formulu vpershe viviv Rolf Nevanlinna Yaksho funkciya f ye golomorfoyu v odinichnomu kruzi z nulyami a1 a2 an rozmishenimi vseredini odinichnogo kruga to dlya kozhnogo z 0 r 0 e i f 0 displaystyle z 0 r 0 e i varphi 0 nbsp v odinichnomu kruzi formula Puassona Yensena maye viglyad log f z 0 k 1 n log z 0 a k 1 a k z 0 1 2 p 0 2 p P r 0 f 0 8 log f e i 8 d 8 displaystyle log f z 0 sum k 1 n log left frac z 0 a k 1 bar a k z 0 right frac 1 2 pi int 0 2 pi P r 0 varphi 0 theta log f e i theta d theta nbsp Tut P r w n Z r n e i n w displaystyle P r omega sum n in mathbb Z r n e in omega nbsp ye yadrom Puassona v odinichnomu kruzi Yaksho funkciya f ne maye nuliv v odinichnomu kruzi to formula Puassona Yensena zvoditsya do log f z 0 1 2 p 0 2 p P r 0 f 0 8 log f e i 8 d 8 displaystyle log f z 0 frac 1 2 pi int 0 2 pi P r 0 varphi 0 theta log f e i theta d theta nbsp tobto do integralnoyi formuli Puassona dlya garmonichnoyi funkciyi log f z displaystyle log f z nbsp Literatura red Greene Robert E Krantz Steven G 2002 Function Theory of One Complex Variable vid 2nd American Mathematical Society ISBN 0 8218 2905 X Otrimano z https uk wikipedia org w index php title Formula Yensena amp oldid 30536457
