Teta funkciyi cili funkciyi kompleksnoyi zminnoyi mozhlivo zalezhni vid dodatkovih parametriv sho ye kvazidvoperiodichnimi tobto krim periodu w displaystyle omega mayut she tak zvanij kvaziperiod w t displaystyle omega tau pri dodavanni yakogo do znachennya argumentu znachennya funkciyi mnozhitsya na deyakij multiplikator Osoblivo velike znachennya mayut teta funkciyi Yakobi chotiri teta funkciyi zalezhni vid parametra t displaystyle tau sho vikoristovuyutsya v teoriyi eliptichnih funkcij modulyarnih form i inshih Zmist 1 Zagalni teta funkciyi 2 Teta funkciyi Yakobi 2 1 Osnovna teta funkciya Yakobi 2 2 Inshi teta funkciyi Yakobi 2 3 Teta konstanti 2 4 Oznachennya za dopomogoyu nomu 2 5 Vlastivosti 2 5 1 Periodichnist i kvaziperiodichnist 2 5 2 Integralni predstavlennya 2 5 3 Nuli teta funkcij Yakobi 2 5 4 Rivnosti Yakobi 2 5 5 Inshi vlastivosti 3 Div takozh 4 LiteraturaZagalni teta funkciyi RedaguvatiTeta funkciyeyu 8 z displaystyle theta z nbsp nazivayetsya funkciya sho zadovolnyaye vlastivosti 8 z w 8 z 8 z w t ϕ z 8 z displaystyle theta z omega theta z theta z omega tau phi z theta z nbsp Yak periodichna cila funkciya 8 z displaystyle theta z nbsp zavzhdi rivna sumi ryadu 8 z n Z c n exp 2 p i n w displaystyle theta z sum n in mathbb Z c n exp left frac 2 pi in omega right nbsp Dani ryadi nazivayutsya teta ryadami Na praktici najvazhlivishimi ye multiplikatori viduϕ z q exp 2 p i k z displaystyle phi z q exp 2 pi ikz nbsp de k naturalne chislo sho nazivayetsya poryadkom abo vagoyu teta funkciyi q chislovij mnozhnik Teta funkciyi Yakobi RedaguvatiOsnovna teta funkciya Yakobi Redaguvati Osnovnoyu teta funkciyeyu Yakobi nazivayetsya funkciya dvoh kompleksnih zminnih sho za oznachennyam rivna ϑ z t n e p i n 2 t 2 p i n z displaystyle vartheta z tau sum n infty infty e pi in 2 tau 2 pi inz nbsp Danij ryad ye normalno zbizhnim na mnozhini C H displaystyle mathbb C times mathbb H nbsp de H z C ℑ z gt 0 displaystyle mathbb H z in mathbb C Im z gt 0 nbsp ye verhnoyu kompleksnoyu napivploshinoyu Dlya vsih t H displaystyle tau in mathbb H nbsp funkciya ϑ t displaystyle vartheta cdot tau nbsp ye ciloyu funkciyeyu dlya vsih z C displaystyle z in mathbb C nbsp funkciya ϑ z displaystyle vartheta z cdot nbsp ye golomorfnoyu na mnozhini H displaystyle mathbb H nbsp Inshi teta funkciyi Yakobi Redaguvati Cherez osnovnu teta funkciyu Yakobi mozhna vvesti she tri teta funkciyi ϑ 0 z t ϑ 0 1 z t ϑ z 1 2 t n 1 n e p i n 2 t 2 p i n z displaystyle vartheta 0 z tau vartheta 0 1 z tau vartheta left z frac 1 2 tau right sum n infty infty 1 n cdot e pi in 2 tau 2 pi inz nbsp ϑ 2 z t ϑ 1 0 z t e p i t 4 p i z ϑ z t 2 t n e p i n 1 2 2 t 2 p i n 1 2 z displaystyle vartheta 2 z tau vartheta 1 0 z tau e pi i frac tau 4 pi iz cdot vartheta left z frac tau 2 tau right sum n infty infty e pi i left n frac 1 2 right 2 tau 2 pi i left n frac 1 2 right z nbsp ϑ 1 z t ϑ 1 1 z t e p i t 4 p i z 1 2 ϑ z t 1 2 t i n 1 n e p i n 1 2 2 t 2 p i n 1 2 z displaystyle vartheta 1 z tau vartheta 1 1 z tau e pi i frac tau 4 pi i left z frac 1 2 right cdot vartheta left z frac tau 1 2 tau right i cdot sum n infty infty 1 n cdot e pi i left n frac 1 2 right 2 tau 2 pi i left n frac 1 2 right z nbsp V cih poznachennyah ϑ z t ϑ 3 z t ϑ 0 0 z t displaystyle vartheta z tau vartheta 3 z tau vartheta 0 0 z tau nbsp Teta konstanti Redaguvati Dlya znachennya z 0 displaystyle z 0 nbsp otrimayemo funkciyi viznacheni na verhnij kompleksnij napivploshini yaki takozh nazivayutsya teta konstantami ϑ t ϑ 0 t n e p i n 2 t 1 2 n 1 e p i n 2 t displaystyle vartheta tau vartheta 0 tau sum n infty infty e pi in 2 tau 1 2 sum n 1 infty e pi in 2 tau nbsp Oznachennya za dopomogoyu nomu Redaguvati Oznachennya teta funkcij mozhna dati ne lishe v terminah zminnih z i t ale i v zminnih w i noma q de w epiz i q epit V cih zminnih funkciyi rivni ϑ 00 w q n w 2 n q n 2 ϑ 01 w q n 1 n w 2 n q n 2 ϑ 10 w q n w 2 n 1 2 q n 1 2 2 ϑ 11 w q i n 1 n w 2 n 1 2 q n 1 2 2 displaystyle begin aligned vartheta 00 w q amp sum n infty infty w 2 n q n 2 quad amp vartheta 01 w q amp sum n infty infty 1 n w 2 n q n 2 3pt vartheta 10 w q amp sum n infty infty w 2 n frac 1 2 q left n frac 1 2 right 2 quad amp vartheta 11 w q amp i sum n infty infty 1 n w 2 n frac 1 2 q left n frac 1 2 right 2 end aligned nbsp Ci fomuli mozhna vikoristati dlya oznachen teta funkcij v polyah de eksponencijne vidobrazhennya mozhe ne buti vsyudi viznachenim napriklad v polyah p adichnih chisel Vlastivosti Redaguvati Periodichnist i kvaziperiodichnist Redaguvati Dlya fiksovanogo t teta funkciyi Yakobi ye periodichnimi z periodami 1 abo 2 i kvaziperiodichnimi shodo periodu t a same dlya vsih z C displaystyle z in mathbb C nbsp vikonuyutsya rivnosti ϑ 00 z 1 t ϑ 00 z t ϑ 00 z t t exp p i t 2 p i z ϑ 00 z t displaystyle vartheta 00 z pm 1 tau vartheta 00 z tau vartheta 00 z pm tau tau exp left pi i tau mp 2 pi iz right vartheta 00 z tau nbsp ϑ 01 z 1 t ϑ 01 z t ϑ 01 z t t exp p i t 2 p i z ϑ 01 z t displaystyle vartheta 01 z pm 1 tau vartheta 01 z tau vartheta 01 z pm tau tau exp left pi i tau mp 2 pi iz right vartheta 01 z tau nbsp ϑ 10 z 1 t ϑ 10 z t ϑ 10 z t t exp p i t 2 p i z ϑ 10 z t displaystyle vartheta 10 z pm 1 tau vartheta 10 z tau vartheta 10 z pm tau tau exp left pi i tau mp 2 pi iz right vartheta 10 z tau nbsp ϑ 11 z 1 t ϑ 11 z t ϑ 11 z t t exp p i t 2 p i z ϑ 11 z t displaystyle vartheta 11 z pm 1 tau vartheta 11 z tau vartheta 11 z pm tau tau exp left pi i tau mp 2 pi iz right vartheta 11 z tau nbsp Tobto dlya fiksovanogo t teta funkciyi Yakobi ye teta funkciyami zgidno oznachennya zagalnoyi teta funkciyi Integralni predstavlennya Redaguvati Dlya teta funkcij Yakobi spravedlivimi ye integralni predstavlennya ϑ 00 z t i i i e i p t u 2 cos 2 u z p u sin p u d u ϑ 01 z t i i i e i p t u 2 cos 2 u z sin p u d u ϑ 10 z t i e i z 1 4 i p t i i e i p t u 2 cos 2 u z p u p t u sin p u d u ϑ 11 z t e i z 1 4 i p t i i e i p t u 2 cos 2 u z p t u sin p u d u displaystyle begin aligned vartheta 00 z tau amp i int i infty i infty e i pi tau u 2 frac cos 2uz pi u sin pi u mathrm d u 6pt vartheta 01 z tau amp i int i infty i infty e i pi tau u 2 frac cos 2uz sin pi u mathrm d u 6pt vartheta 10 z tau amp ie iz frac 1 4 i pi tau int i infty i infty e i pi tau u 2 frac cos 2uz pi u pi tau u sin pi u mathrm d u 6pt vartheta 11 z tau amp e iz frac 1 4 i pi tau int i infty i infty e i pi tau u 2 frac cos 2uz pi tau u sin pi u mathrm d u end aligned nbsp Nuli teta funkcij Yakobi Redaguvati Dlya fiksovanogo t teta funkciyi Yakobi ϑ 00 z t 0 z m n t 1 2 t 2 ϑ 11 z t 0 z m n t ϑ 10 z t 0 z m n t 1 2 ϑ 01 z t 0 z m n t t 2 displaystyle begin aligned vartheta 00 z tau amp 0 quad amp Longleftrightarrow amp amp quad z amp m n tau frac 1 2 frac tau 2 3pt vartheta 11 z tau amp 0 quad amp Longleftrightarrow amp amp quad z amp m n tau 3pt vartheta 10 z tau amp 0 quad amp Longleftrightarrow amp amp quad z amp m n tau frac 1 2 3pt vartheta 01 z tau amp 0 quad amp Longleftrightarrow amp amp quad z amp m n tau frac tau 2 end aligned nbsp de m n dovilni cili chisla Rivnosti Yakobi Redaguvati Rivnosti Yakobi viznachayut povedinku teta funkcij Yakobi pid vplivom diyi modulyarnoyi grupi porodzhenoyi peretvorennyami t t 1 i t 1 t Rivnyannya dlya pershogo peretvorennya utvoryuyutsya z vrahuvannyam togo sho dodavannya 1 do t maye takij zhe efekt na znachennya funkciyi yak dodavannya 1 2 do z Dlya viznachennya vplivu drugogo peretvorennya poznachimo a i t 1 2 exp p t i z 2 displaystyle alpha i tau frac 1 2 exp left frac pi tau iz 2 right nbsp Todi ϑ 00 z t 1 t a ϑ 00 z t ϑ 01 z t 1 t a ϑ 10 z t ϑ 10 z t 1 t a ϑ 01 z t ϑ 11 z t 1 t i a ϑ 11 z t displaystyle begin aligned vartheta 00 left frac z tau frac 1 tau right amp alpha vartheta 00 z tau quad amp vartheta 01 left frac z tau frac 1 tau right amp alpha vartheta 10 z tau 3pt vartheta 10 left frac z tau frac 1 tau right amp alpha vartheta 01 z tau quad amp vartheta 11 left frac z tau frac 1 tau right amp i alpha vartheta 11 z tau end aligned nbsp Inshi vlastivosti Redaguvati Formula dobutku ϑ 00 z t m 1 1 exp 2 m p i t 1 exp 2 m 1 p i t 2 p i z 1 exp 2 m 1 p i t 2 p i z displaystyle vartheta 00 z tau prod m 1 infty big 1 exp 2m pi i tau big Big 1 exp big 2m 1 pi i tau 2 pi iz big Big Big 1 exp big 2m 1 pi i tau 2 pi iz big Big nbsp Vsi teta funkciyi Yakobi zadovolnyayut diferencialnomu rivnyannyu t ϑ x i t 1 4 p 2 x 2 ϑ x i t displaystyle frac partial partial t vartheta x it frac 1 4 pi frac partial 2 partial x 2 vartheta x it nbsp Zv yazok z eliptichnoyu funkciyeyu Vejyershtrasa z t log ϑ 11 z t c displaystyle wp z tau big log vartheta 11 z tau big c nbsp de pohidni ye shodo zminnoyi z i konstanta c vibirayetsya tak shob rozklad z v ryad Lorana v tochci z 0 mav nulovij dodanok nulovogo stepenya Zv yazok z dzeta funkciyeyu Rimana G s 2 p s 2 z s 1 2 0 ϑ 00 0 i t 1 t s 2 d t t displaystyle Gamma left frac s 2 right pi frac s 2 zeta s frac 1 2 int 0 infty big vartheta 00 0 it 1 big t frac s 2 frac mathrm d t t nbsp Zv yazok z eta funkciyeyu Dedekinda Nehaj h t eta funkciyeyu Dedekinda Todi ϑ 10 0 t 2 h 2 2 t h t ϑ 00 0 t h 5 t h 2 1 2 t h 2 2 t h 2 1 2 t 1 h t 1 ϑ 01 0 t h 2 1 2 t h t displaystyle begin aligned vartheta 10 0 tau amp frac 2 eta 2 2 tau eta tau 3pt vartheta 00 0 tau amp frac eta 5 tau eta 2 left frac 1 2 tau right eta 2 2 tau frac eta 2 left frac 1 2 tau 1 right eta tau 1 3pt vartheta 01 0 tau amp frac eta 2 left frac 1 2 tau right eta tau end aligned nbsp i ϑ 10 0 t ϑ 00 0 t ϑ 01 0 t 2 h 3 t displaystyle vartheta 10 0 tau vartheta 00 0 tau vartheta 01 0 tau 2 eta 3 tau nbsp Div takozh RedaguvatiEliptichna funkciya Eliptichni funkciyi YakobiLiteratura RedaguvatiAbramowitz Milton Stegun Irene A 1964 Handbook of Mathematical Functions New York Dover Publications sec 16 27ff ISBN 0 486 61272 4 Akhiezer Naum Illyich 1990 1970 Elements of the Theory of Elliptic Functions AMS Translations of Mathematical Monographs 79 Providence RI AMS ISBN 0 8218 4532 2 Bellman Richard 1961 A Brief Introduction to Theta Functions Selected Topics in Mathematics New York Holt Rinehart and Winston Hardy G H Wright E M 1959 An Introduction to the Theory of Numbers vid 4th Oxford Clarendon Press Mumford David 1983 Tata Lectures on Theta I Boston Birkhauser ISBN 3 7643 3109 7 Rauch Harry E Farkas Hershel M 1974 Theta Functions with Applications to Riemann Surfaces Baltimore Williams amp Wilkins ISBN 0 683 07196 3 Whittaker E T Watson G N 1927 A Course in Modern Analysis vid 4th Cambridge Cambridge University Press ch 21 Otrimano z https uk wikipedia org w index php title Teta funkciya amp oldid 35035557
