Eliptichni funkciyi Yakobi nabir osnovnih eliptichnih funkcij kompleksnoyi zminnoyi i dopomizhnih teta funkcij yaki mayut velike istorichne znachennya i pryame vidnoshennya do deyakih prikladnih zadach napriklad rivnyannya mayatnika Voni takozh mayut korisni analogiyi z trigonometrichnimi funkciyami yak pokazuye vidpovidne poznachennya s n displaystyle operatorname mathrm sn dlya sin displaystyle sin Voni ne dayut najprostishij sposib rozvinuti zagalnu teoriyu eliptichnih funkcij tomu v u vstupnih knigah voni mensh populyarni nizh eliptichni funkciyi Vejyershtrasa Eliptichni funkciyi Yakobi mayut v osnovnomu paralelogrami po dva prostih polyusi i dva prostih nulya Zmist 1 Oznachennya 1 1 Yak meromorfni funkciyi 1 2 Yak oberneni funkciyi do eliptichnih integraliv 1 3 Oznachennya v terminah teta funkcij 2 Poznachennya 3 Inshi funkciyi 4 Dodatkovi teoremi 5 Trigonometrichni i giperbolichni funkciyi yak okremij vipadok eliptichnih 6 Spivvidnoshennya mizh kvadratami funkcij 7 Nom 8 Rozv yazki nelinijnih zvichajnih diferencialnih rivnyan 9 Div takozh 10 Posilannya 11 LiteraturaOznachennya RedaguvatiYak meromorfni funkciyi Redaguvati nbsp Paralelogram Yakobi Funkciyi Yakobi ye eliptichnimi funkciyami tobto podvijno periodichnimi meromorfnimi funkciyami kompleksnoyi zminnoyi Tobto faktichno yih znachennya viznachayetsya na tori abo osnovnomu paralelogrami Yaksho cya funkciya ye vsyudi golomorfnoyu to zgidno z teoremoyu Liuvilya vona bude konstantoyu Z vlastivostej lishkiv ta podvijnoyi periodichnosti viplivaye takozh sho eliptichni funkciyi ne mozhut v osnovnomu paralelogrami mati yedinogo polyusa poryadku 1 Vidpovidno najprostishimi nestalimi funkciyami ye funkciyi z yedinim polyusom poryadku dva i dvoma polyusami poryadku 1 Pershimi ye eliptichni funkciyi Vejyershtrasa drugimi eliptichni funkciyi Yakobi Zagalom isnuye 12 principovo vidminnih eliptichnih funkcij Yakobi Zagalom voni zalezhat vid osnovnogo paralelograma Nehaj viznacheno paralelogram yakij ne bude osnovnim yak na malyunku z vershinami 0 K K iK iK sho dlya zruchnosti notaciyi poznacheni yak s c d i n vidpovidno Dijsni chisla K i K nazivayutsya chvertyami periodiv 12 funkcij poznachayutsya sc sd sn cd cn cs dn ds dc ns nc i nd Voni ye yedinimi eliptichnimi funkciyami sho zadovolnyayut umovi Funkciya maye prostij nul v kuti p viznachenogo paralelograma i prostij polyus v kuti q V inshih dvoh kutah polyusiv i nuliv nemaye Vidstan vid p do q ye polovinoyu periodu funkciyi pq u tobto funkciya pq u ye periodichnoyu v napryamku pq z periodom vdvichi bilshim nizh vidstan vid p do q Vidstani vid p do inshih tochok ye chvertyami periodiv Rozklad funkciyi pq u v ryad Tejlora shodo u v okoli tochki p maye chlenom najmenshogo stepenya u chlenom najmenshogo stepenya pri rozkladi v ryad Lorana v okoli q ye 1 u v inshih kutah rozklad v ryad Tejlora pochinayetsya z 1 Napriklad funkciya dn maye nul v tochci d i polyus v tochci n Vona periodichna z periodami 2K i 4iK Yak oberneni funkciyi do eliptichnih integraliv Redaguvati Navedene vishe oznachennya v terminah meromorfnih funkcij ye dosit abstraktnim Isnuye bilsh proste ale absolyutno ekvivalentne oznachennya sho zadaye eliptichni funkciyi yak zvorotni do nepovnogo eliptichnomu integralu pershogo rodu nehaj u 0 ϕ d 8 1 m sin 2 8 displaystyle u int limits 0 phi frac d theta sqrt 1 m sin 2 theta nbsp Eliptichna funkciya s n u displaystyle sn u nbsp zadayetsya yak sn u sin ϕ displaystyle operatorname sn u sin phi nbsp i c n u displaystyle cn u nbsp viznachayetsya cn u cos ϕ displaystyle operatorname cn u cos phi nbsp a dn u 1 m sin 2 ϕ displaystyle operatorname dn u sqrt 1 m sin 2 phi nbsp Tut kut ϕ displaystyle phi nbsp nazivayetsya amplitudoyu dn u D u displaystyle operatorname dn u Delta u nbsp nazivayetsya delta amplitudoyu Znachennya m ye vilnim parametrom yakij ye dijsnim chislom v diapazoni 0 m 1 displaystyle 0 leq m leq 1 nbsp i takim chinom eliptichni funkciyi ye funkciyami dvoh argumentiv amplitudi ϕ displaystyle phi nbsp i parametra m Reshta dev yat eliptichnih funkcij legko pobuduvati z troh vishenavedenih Ce bude zrobleno nizhche Koli ϕ p 2 displaystyle phi pi 2 nbsp to u dorivnyuye chverti periodu K Oznachennya v terminah teta funkcij Redaguvati Ekvivalentno eliptichni funkciyi Yakobi mozhna viznachiti v terminah teta funkcij Yaksho mi viznachimo ϑ 0 t displaystyle vartheta 0 tau nbsp yak ϑ displaystyle vartheta nbsp i ϑ 01 0 t ϑ 10 0 t ϑ 11 0 t displaystyle vartheta 01 0 tau vartheta 10 0 tau vartheta 11 0 tau nbsp vidpovidno yak ϑ 01 ϑ 10 ϑ 11 displaystyle vartheta 01 vartheta 10 vartheta 11 nbsp teta konstanti todi eliptichnij modul k dorivnyuye k ϑ 10 ϑ 2 displaystyle k left vartheta 10 over vartheta right 2 nbsp Vvazhayuchi u p ϑ 2 z displaystyle u pi vartheta 2 z nbsp otrimayemo sn u k ϑ ϑ 11 z t ϑ 10 ϑ 01 z t cn u k ϑ 01 ϑ 10 z t ϑ 10 ϑ 01 z t dn u k ϑ 01 ϑ z t ϑ ϑ 01 z t displaystyle begin aligned operatorname sn u k amp vartheta vartheta 11 z tau over vartheta 10 vartheta 01 z tau 7pt operatorname cn u k amp vartheta 01 vartheta 10 z tau over vartheta 10 vartheta 01 z tau 7pt operatorname dn u k amp vartheta 01 vartheta z tau over vartheta vartheta 01 z tau end aligned nbsp Oskilki funkciyi Yakobi viznachayutsya v terminah eliptichnogo modulya k t displaystyle k tau nbsp neobhidno znajti oberneni do nih i zapisati t v terminah k Pochnemo z dodatkovogo modulya k 1 k 2 displaystyle k sqrt 1 k 2 nbsp Yak funkciya vid t vin rivnij k t ϑ 01 ϑ 2 displaystyle k tau vartheta 01 over vartheta 2 nbsp Vvedemo poznachennya ℓ 1 2 1 k 1 k 1 2 ϑ ϑ 01 ϑ ϑ 01 displaystyle ell 1 over 2 1 sqrt k over 1 sqrt k 1 over 2 vartheta vartheta 01 over vartheta vartheta 01 nbsp Viznachimo takozh nom q yak q exp p i t displaystyle q exp pi i tau nbsp i rozklademo ℓ displaystyle ell nbsp v ryad za stepenyami noma q otrimayemo ℓ q q 9 q 25 1 2 q 4 2 q 16 displaystyle ell q q 9 q 25 cdots over 1 2q 4 2q 16 cdots nbsp Mozhna zapisati rozklad v ryad q ℓ 2 ℓ 5 15 ℓ 9 150 ℓ 13 1707 ℓ 17 20910 ℓ 21 268616 ℓ 25 displaystyle q ell 2 ell 5 15 ell 9 150 ell 13 1707 ell 17 20910 ell 21 268616 ell 25 cdots nbsp Oskilki mi mozhemo rozglyanuti okremij vipadok koli uyavna chastina t bilsha abo rivna 3 2 displaystyle sqrt 3 2 nbsp mi mozhemo skazati sho znachennya q menshe abo rivne exp p 3 2 displaystyle exp pi sqrt 3 2 nbsp Dlya takih malih znachen vishenavedenij ryad zbigayetsya duzhe shvidko i ce dozvolyaye legko znajti vidpovidne znachennya dlya q Poznachennya RedaguvatiDlya eliptichnih funkcij mozhna zustriti riznomanitni poznachennya Eliptichni funkciyi funkciyi dvoh zminnih Pershu zminnu mozhna dati v terminah amplitudi f abo zazvichaj v terminah u yak nizhche Drugu zminnu mozhna bulo b dati v terminah parametra m abo yak eliptichnij modul k de k 2 m displaystyle k 2 m nbsp abo v terminah modulyarnogo kuta o e displaystyle o varepsilon nbsp de m sin 2 o e displaystyle m sin 2 o varepsilon nbsp Inshi funkciyi RedaguvatiZminoyu dvoh bukv v nazvi funkcij zazvichaj poznachayut oberneni funkciyi do troh osnovnih funkcij navedenih vishe ns u 1 sn u nc u 1 cn u nd u 1 dn u displaystyle begin aligned operatorname ns u amp frac 1 operatorname sn u 8pt operatorname nc u amp frac 1 operatorname cn u 8pt operatorname nd u amp frac 1 operatorname dn u end aligned nbsp Chastki troh golovnih funkcij poznachayut pershoyu literoyu chiselnika i pershoyu literoyu znamennika sc u sn u cn u sd u sn u dn u dc u dn u cn u ds u dn u sn u cs u cn u sn u cd u cn u dn u displaystyle begin aligned operatorname sc u amp frac operatorname sn u operatorname cn u 8pt operatorname sd u amp frac operatorname sn u operatorname dn u 8pt operatorname dc u amp frac operatorname dn u operatorname cn u 8pt operatorname ds u amp frac operatorname dn u operatorname sn u 8pt operatorname cs u amp frac operatorname cn u operatorname sn u 8pt operatorname cd u amp frac operatorname cn u operatorname dn u end aligned nbsp Dlya krashogo zapam yatovuvannya bilsh korotko mozhna zapisati pq u pr u q r u displaystyle operatorname pq u frac operatorname pr u operatorname qr u nbsp de vsi bukvi p q i r ye bud yakimi bukvami s c d n slid pam yatati sho ss cc dd nn 1 Dodatkovi teoremi RedaguvatiFunkciyi zadovolnyayut dvom algebrayichnim spivvidnoshennyam cn 2 u k sn 2 u k 1 displaystyle operatorname cn 2 u k operatorname sn 2 u k 1 nbsp dn 2 u k k 2 sn 2 u k 1 displaystyle operatorname dn 2 u k k 2 operatorname sn 2 u k 1 nbsp Z cogo vidno sho cn sn dn parametrizuyut eliptichnu krivu yaka ye peretinom dvoh kvadrik zadanih vishezaznachenimi dvoma rivnyannyami Na cij krivij mozhna viznachiti grupovij zakon dlya tochok za dopomogoyu dodatkovih formul dlya funkcij Yakobi cn x y cn x cn y sn x sn y dn x dn y 1 k 2 sn 2 x sn 2 y sn x y sn x cn y dn y sn y cn x dn x 1 k 2 sn 2 x sn 2 y dn x y dn x dn y k 2 sn x sn y cn x cn y 1 k 2 sn 2 x sn 2 y displaystyle begin aligned operatorname cn x y amp operatorname cn x operatorname cn y operatorname sn x operatorname sn y operatorname dn x operatorname dn y over 1 k 2 operatorname sn 2 x operatorname sn 2 y 8pt operatorname sn x y amp operatorname sn x operatorname cn y operatorname dn y operatorname sn y operatorname cn x operatorname dn x over 1 k 2 operatorname sn 2 x operatorname sn 2 y 8pt operatorname dn x y amp operatorname dn x operatorname dn y k 2 operatorname sn x operatorname sn y operatorname cn x operatorname cn y over 1 k 2 operatorname sn 2 x operatorname sn 2 y end aligned nbsp Trigonometrichni i giperbolichni funkciyi yak okremij vipadok eliptichnih RedaguvatiYaksho m 1 to u 0 f d 8 1 sin 2 8 ln 1 cos f tg f displaystyle u int limits 0 varphi frac d theta sqrt 1 sin 2 theta operatorname ln left frac 1 cos varphi operatorname tg varphi right nbsp Zvidsi sin f sn u e u 1 e u 1 th u displaystyle sin varphi operatorname sn u frac e u 1 e u 1 operatorname th u nbsp dd Zvidsi cn u 1 sn 2 u 1 ch u displaystyle operatorname cn u sqrt 1 operatorname sn 2 u frac 1 operatorname ch u nbsp dd i dn u 1 sn 2 u 1 ch u displaystyle operatorname dn u sqrt 1 operatorname sn 2 u frac 1 operatorname ch u nbsp dd Takim chinom pri m 1 eliptichni funkciyi virodzhuyutsya v giperbolichni Yaksho m 0 to u 0 f d 8 f displaystyle u int limits 0 varphi d theta varphi nbsp Zvidsi sin f sin u sn u displaystyle sin varphi sin u operatorname sn u nbsp dd a takozh cn u cos u displaystyle operatorname cn u cos u nbsp dn u 1 displaystyle operatorname dn u 1 nbsp dd Takim chinom pri m 0 eliptichni funkciyi virodzhuyutsya v trigonometrichni Spivvidnoshennya mizh kvadratami funkcij RedaguvatiDlya kvadrativ cih funkcij virni nastupni spivvidnoshennya dn 2 u m 1 m cn 2 u m sn 2 u m displaystyle operatorname dn 2 u m 1 m operatorname cn 2 u m operatorname sn 2 u m nbsp m 1 nd 2 u m 1 m m 1 sd 2 u m cd 2 u m displaystyle m 1 operatorname nd 2 u m 1 mm 1 operatorname sd 2 u m operatorname cd 2 u m nbsp m 1 sc 2 u m 1 m 1 nc 2 u dc 2 u m displaystyle m 1 operatorname sc 2 u m 1 m 1 operatorname nc 2 u operatorname dc 2 u m nbsp cs 2 u m 1 ds 2 u ns 2 u m displaystyle operatorname cs 2 u m 1 operatorname ds 2 u operatorname ns 2 u m nbsp de m m 1 1 displaystyle m m 1 1 nbsp i m k 2 displaystyle m k 2 nbsp Dodatkovi rivnosti dlya kvadrativ mozhna otrimati yaksho zauvazhiti sho pq 2 qp 2 1 displaystyle operatorname pq 2 cdot operatorname qp 2 1 nbsp a takozh pq pr qr displaystyle operatorname pq operatorname pr operatorname qr nbsp de p q r bud yaki literi s c d n i ss cc dd nn 1 Nom RedaguvatiNehaj nom dorivnyuye q exp p K K displaystyle q exp pi K K nbsp i nehaj argument v p u 2 K displaystyle v pi u 2K nbsp Todi funkciyi mozhna predstaviti u viglyadi sum Lamberta sn u 2 p K m n 0 q n 1 2 1 q 2 n 1 sin 2 n 1 v displaystyle operatorname sn u frac 2 pi K sqrt m sum n 0 infty frac q n 1 2 1 q 2n 1 sin 2n 1 v nbsp cn u 2 p K m n 0 q n 1 2 1 q 2 n 1 cos 2 n 1 v displaystyle operatorname cn u frac 2 pi K sqrt m sum n 0 infty frac q n 1 2 1 q 2n 1 cos 2n 1 v nbsp dn u p 2 K 2 p K n 1 q n 1 q 2 n cos 2 n v displaystyle operatorname dn u frac pi 2K frac 2 pi K sum n 1 infty frac q n 1 q 2n cos 2nv nbsp Rozv yazki nelinijnih zvichajnih diferencialnih rivnyan RedaguvatiPohidni troh osnovnih eliptichnih funkcij Yakobi zapisuyutsya u viglyadi d d z s n z k c n z k d n z k displaystyle frac mathrm d mathrm d z mathrm sn z k mathrm cn z k mathrm dn z k nbsp d d z c n z k s n z k d n z k displaystyle frac mathrm d mathrm d z mathrm cn z k mathrm sn z k mathrm dn z k nbsp d d z d n z k k 2 s n z k c n z k displaystyle frac mathrm d mathrm d z mathrm dn z k k 2 mathrm sn z k mathrm cn z k nbsp Vikoristovuyuchi teoremu formulyuvannya yakoyi navedena vishe otrimayemo dlya zadanogo k 0 lt k lt 1 rivnyannya rozv yazkami yakih ye eliptichni funkciyi Yakobi s n x k displaystyle mathrm sn x k nbsp ye rozv yazkom rivnyand 2 y d x 2 1 k 2 y 2 k 2 y 3 0 displaystyle frac mathrm d 2 y mathrm d x 2 1 k 2 y 2k 2 y 3 0 nbsp dd i d y d x 2 1 y 2 1 k 2 y 2 displaystyle left frac mathrm d y mathrm d x right 2 1 y 2 1 k 2 y 2 nbsp dd c n x k displaystyle mathrm cn x k nbsp ye rozv yazkom rivnyand 2 y d x 2 1 2 k 2 y 2 k 2 y 3 0 displaystyle frac mathrm d 2 y mathrm d x 2 1 2k 2 y 2k 2 y 3 0 nbsp dd i d y d x 2 1 y 2 1 k 2 k 2 y 2 displaystyle left frac mathrm d y mathrm d x right 2 1 y 2 1 k 2 k 2 y 2 nbsp dd d n x k displaystyle mathrm dn x k nbsp ye rozv yazkom rivnyannyad 2 y d x 2 2 k 2 y 2 y 3 0 displaystyle frac mathrm d 2 y mathrm d x 2 2 k 2 y 2y 3 0 nbsp dd i d y d x 2 y 2 1 1 k 2 y 2 displaystyle left frac mathrm d y mathrm d x right 2 y 2 1 1 k 2 y 2 nbsp dd Div takozh RedaguvatiEliptichnij integral Eliptichna funkciya Eliptichni funkciyi VejyershtrassaPosilannya RedaguvatiHazewinkel Michiel red 2001 Jacobi elliptic functions Matematichna enciklopediya Springer ISBN 978 1 55608 010 4 Weisstein Eric W Jacobi Elliptic Functions angl na sajti Wolfram MathWorld Literatura RedaguvatiAbramowitz Milton Stegun Irene A eds 1972 Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables New York Dover ISBN 0 486 61272 4 accessdate vimagaye url dovidka See Chapter 16 Arhivovano 9 veresnya 2009 u Wayback Machine N I Ahiezer 1970 Elementy teorii ellipticheskih funkcij Moskva Nauka Dzh N Vatson E T Uitteker 1963 Kurs sovremennogo analiza Ch 2 Transcendentnye funkcii Moskva Mir ili Moskva URSS 2010 Grigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr Otrimano z https uk wikipedia org w index php title Eliptichni funkciyi Yakobi amp oldid 40618359
