Ortogonalni polinomiErmitaVidkriti Sharlem Ermitom v 1864 rociFormula H n x 1 n e x 2 2 d n d x n e x 2 2 displaystyle H n x 1 n e frac x 2 2 frac d n dx n left e frac x 2 2 right Diferencialne rivnyannya y x x y x n y x 0 displaystyle y x xy x ny x 0 Viznacheni na displaystyle infty infty Vaga e x 2 2 displaystyle e x 2 2 Norma n 2 p displaystyle sqrt n sqrt 2 pi Primitki V fizici chasto vikoristovuyutsya polinomi Ermita viznacheni yakH n x 1 n e x 2 d n d x n e x 2 displaystyle H n x 1 n e x 2 frac d n dx n left e x 2 right H n x 2 n 2 H n 2 x displaystyle H n x 2 frac n 2 H n sqrt 2 x Polinomi Ermita angl Hermite polynomials ortogonalni polinomi sho vikoristovuyutsya v teoriyi jmovirnostej matematichnij fizici pri rozv yazku rivnyannya difuziyi chiselnomu analizi ta kvantovij mehanici yak vlasni funkciyi kvantovogo garmonichnogo oscilyatora Nazvani na chest francuzkogo matematika Sharlya Ermita yakij vviv 1 yih v 1864 roci Zmist 1 Viznachennya 2 Vlastivosti 2 1 Formula dodavannya 3 Diferenciyuvannya ta rekurentni spivvidnoshennya 4 Ortogonalnist 5 Povnota 6 Diferencialni rivnyannya 7 Predstavlennya 8 Zv yazok z inshimi specialnimi funkciyami 9 Zastosuvannya 10 Primitki 11 Literatura 12 Zovnishni posilannyaViznachennya Redaguvati nbsp Grafiki polinomiv Ermita poryadku n 0 1 5 displaystyle n 0 1 5 nbsp Polinomami Ermita nazivayetsya poslidovnist polinomiv H n x displaystyle H n x nbsp n 0 1 displaystyle n 0 1 nbsp sho zadovolnyayut spivvidnoshennyu e t x t 2 2 n 0 H n x t n n displaystyle e tx frac t 2 2 sum n 0 infty H n x frac t n n nbsp z yakogo viplivayeH n x 1 n e x 2 2 d n d x n e x 2 2 displaystyle H n x 1 n e frac x 2 2 frac d n dx n left e frac x 2 2 right nbsp Take oznachennya zdebilshogo vikoristovuyetsya v teoriyi jmovirnostej U fizici zdebilshogo v kvantovij mehanici vikoristovuyut nastupne oznachennya H n x 1 n e x 2 d n d x n e x 2 displaystyle H n x 1 n e x 2 frac d n dx n left e x 2 right nbsp Zv yazok mizh fizichnimi ta jmovirnisnimi polinomami Ermita zdijsnyuyetsya cherez nastupne rivnyannya H n x 2 n 2 H n 2 x displaystyle H n x 2 frac n 2 H n sqrt 2 x nbsp V cij statti budut vikoristovuvatisya jmovirnisni polinomi yaksho ne zaznacheno inshe Yavni virazi pershih odinadcyati polinomiv Ermita mayut takij viglyad H 0 x 1 displaystyle H 0 x 1 nbsp H 1 x x displaystyle H 1 x x nbsp H 2 x x 2 1 displaystyle H 2 x x 2 1 nbsp H 3 x x 3 3 x displaystyle H 3 x x 3 3x nbsp H 4 x x 4 6 x 2 3 displaystyle H 4 x x 4 6x 2 3 nbsp H 5 x x 5 10 x 3 15 x displaystyle H 5 x x 5 10x 3 15x nbsp H 6 x x 6 15 x 4 45 x 2 15 displaystyle H 6 x x 6 15x 4 45x 2 15 nbsp H 7 x x 7 21 x 5 105 x 3 105 x displaystyle H 7 x x 7 21x 5 105x 3 105x nbsp H 8 x x 8 28 x 6 210 x 4 420 x 2 105 displaystyle H 8 x x 8 28x 6 210x 4 420x 2 105 nbsp H 9 x x 9 36 x 7 378 x 5 1260 x 3 945 x displaystyle H 9 x x 9 36x 7 378x 5 1260x 3 945x nbsp H 10 x x 10 45 x 8 630 x 6 3150 x 4 4725 x 2 945 displaystyle H 10 x x 10 45x 8 630x 6 3150x 4 4725x 2 945 nbsp Zagalnij viraz dlya polinomiv Ermita maye viglyad H n x j 0 n 2 1 j 2 j n j n 2 j x n 2 j x n n n 1 2 x n 2 1 4 n n 1 n 2 n 3 2 x n 4 displaystyle H n x sum j 0 n 2 frac 1 j 2 j frac n j n 2j x n 2j x n frac n n 1 2 x n 2 frac 1 4 frac n n 1 n 2 n 3 2 x n 4 ldots nbsp Vlastivosti RedaguvatiPolinom H n x displaystyle H n x nbsp mistit chleni lishe tiyeyi zh parnosti sho j same chislo n displaystyle n nbsp H 2 n x H 2 n x H 2 n 1 x H 2 n 1 x n 0 1 2 displaystyle H 2n x H 2n x H 2n 1 x H 2n 1 x n 0 1 2 ldots nbsp Pri x 0 displaystyle x 0 nbsp mayut misce taki spivvidnoshennya H 2 n 0 1 n 2 n 2 n n H 2 n 1 0 n 0 1 2 displaystyle H 2n 0 frac 1 n 2 n frac 2n n H 2n 1 0 n 0 1 2 ldots nbsp Rivnyannya H n x 0 displaystyle H n x 0 nbsp maye n displaystyle n nbsp dijsnih koreniv sho ye poparno simetrichnimi vidnosno pochatku sistemi koordinat i modul zhodnogo z nih ne perevishuye velichini n n 1 2 displaystyle sqrt n n 1 2 nbsp Koreni polinoma H n x 0 displaystyle H n x 0 nbsp cherguyutsya z korenyami polinoma H n 1 x 0 displaystyle H n 1 x 0 nbsp Polinom H n x displaystyle H n x nbsp mozhna predstaviti u viglyadi viznachnika matrici n n displaystyle n times n nbsp H n x x n 1 0 0 0 1 x n 2 0 0 0 1 x n 3 0 0 0 0 0 x displaystyle H n x left begin array cccccc x amp n 1 amp 0 amp 0 amp cdots amp 0 1 amp x amp n 2 amp 0 amp cdots amp 0 0 amp 1 amp x amp n 3 amp cdots amp 0 cdots amp cdots amp cdots amp cdots amp cdots amp cdots 0 amp 0 amp 0 amp 0 amp cdots amp x end array right nbsp Formula dodavannya Redaguvati Maye misce nastupna formula dodavannya polinomiv Ermita a 1 2 a 2 2 a n 2 m 2 m H m a 1 x 1 a 2 x 2 a n x n a 1 2 a 2 2 a n 2 m 1 m n m a 1 m 1 m 1 a n m n m n H m 1 x 1 H m n x n displaystyle frac a 1 2 a 2 2 cdots a n 2 frac mu 2 mu H mu left frac a 1 x 1 a 2 x 2 cdots a n x n sqrt a 1 2 a 2 2 cdots a n 2 right sum m 1 cdots m n mu frac a 1 m 1 m 1 cdots frac a n m n m n H m 1 x 1 cdots H m n x n nbsp Chastkovimi vipadkami takoyi formuli ye taki a 1 a 2 a n 1 displaystyle a 1 a 2 cdots a n 1 nbsp x 1 x 2 x n displaystyle x 1 x 2 cdots x n nbsp Todin m 2 H m n x m 1 m n m m m 1 m n H m 1 x H m n x displaystyle n frac mu 2 H mu sqrt n x sum m 1 cdots m n mu frac mu m 1 cdots m n H m 1 x cdots H m n x nbsp n 2 displaystyle n 2 nbsp a 1 a 2 1 displaystyle a 1 a 2 1 nbsp x 1 2 x x 2 2 y displaystyle x 1 sqrt 2 x x 2 sqrt 2 y nbsp Todi2 m H m x y p q r s m m p q r s H p x H q x H r x H s x displaystyle 2 mu H mu x y sum p q r s mu frac mu p q r s H p x H q x H r x H s x nbsp Diferenciyuvannya ta rekurentni spivvidnoshennya RedaguvatiPohidna k displaystyle k nbsp go poryadku vid polinoma Ermita H n x displaystyle H n x nbsp n k displaystyle n geq k nbsp takozh ye polinomom Ermita d k d x k H n x n n 1 n k 1 H n k x displaystyle frac d k dx k H n x n n 1 cdots n k 1 H n k x nbsp zvidki viplivaye spivvidoshennya dlya pershoyi pohidnoyiH n x d H n x d x n H n 1 x displaystyle H n x frac dH n x dx nH n 1 x nbsp ta rekurentne spivvidnoshennya mizh troma poslidovnimi polinomami H n x x H n 1 x n 1 H n 2 x 0 n 2 displaystyle H n x xH n 1 x n 1 H n 2 x 0 n geq 2 nbsp Ortogonalnist RedaguvatiPolinomi Ermita utvoryuyut povnu ortogonalnu sistemu na intervali displaystyle infty infty nbsp z vagoyu e x 2 2 displaystyle e x 2 2 nbsp H n x H m x e x 2 2 d x n 2 p d n m displaystyle int infty infty H n x H m x e x 2 2 dx n sqrt 2 pi delta mathit nm nbsp de d m n displaystyle delta mn nbsp delta simvol Kronekera Vazhlivim naslidkom ortogonalnosti polinomiv Ermita ye mozhlivist rozkladu riznih funkcij v ryadi po polinomah Ermita Dlya bud yakogo nevid yemnogo cilogo p displaystyle p nbsp spravedlivij zapisx p p k 0 k p 2 1 2 k 1 k p 2 k H p 2 k x displaystyle frac x p p sum k 0 k leq p 2 frac 1 2 k frac 1 k p 2k H p 2k x nbsp Z nogo viplivaye zv yazok mizh koeficiyentami rozkladu funkciyi v ryad Maklorena f x n 0 a n x n displaystyle f x sum n 0 infty a n x n nbsp ta koeficiyentami rozkladu ciyeyi zh funkciyi po polinomah Ermita f x n 0 A n H n x displaystyle f x sum n 0 infty A n H n x nbsp sho nosyat nazvu spivvidnoshen Nilsa Nilsona A n 1 n k 0 1 2 k n 2 k k a n 2 k a n 1 n k 0 1 k 2 k n 2 k k A n 2 k displaystyle A n frac 1 n sum k 0 infty frac 1 2 k frac n 2k k a n 2k a n frac 1 n sum k 0 infty frac 1 k 2 k frac n 2k k A n 2k nbsp Napriklad rozklad funkciyi Kummera matime takij viglyad 1 F 1 a g x n 0 a n g n 1 n 2 F 2 a n 2 a n 1 2 g n 2 g n 1 2 1 2 H n x a b G a b G a displaystyle 1 F 1 alpha gamma x sum n 0 infty frac alpha n gamma n 1 n 2 F 2 left frac alpha n 2 frac alpha n 1 2 frac gamma n 2 frac gamma n 1 2 frac 1 2 right H n x a b equiv frac Gamma a b Gamma a nbsp de 2 F 2 a 1 a 2 b 1 b 2 x displaystyle 2 F 2 a 1 a 2 b 1 b 2 x nbsp uzagalnena gipergeometrichna funkciya drugogo poryadku G x displaystyle Gamma x nbsp gamma funkciya Rozklad funkcij sho mistyat eksponentu Dlya bud yakoyi funkciyi sho zapisuyetsya yak superpoziciya eksponent f x k 1 p c k e a k x displaystyle f x sum k 1 p c k e alpha k x nbsp mozhna zapisati nastupnij rozklad po polinomah Ermita f x n 0 A n H n x A n 1 n k 1 p c k a k n e a k 2 2 displaystyle f x sum n 0 infty A n H n x A n frac 1 n sum k 1 p c k alpha k n e frac alpha k 2 2 nbsp Zokrema rozkladi vidomih giperbolichnih ta trigonometrichnih funkcij mayut viglyad cosh t x e t 2 2 n 0 t 2 n 2 n H 2 n x sinh t x e t 2 2 n 0 t 2 n 1 2 n 1 H 2 n 1 x displaystyle cosh tx e frac t 2 2 sum n 0 infty frac t 2n 2n H 2n x sinh tx e frac t 2 2 sum n 0 infty frac t 2n 1 2n 1 H 2n 1 x nbsp cos t x e t 2 2 n 0 1 n t 2 n 2 n H 2 n x sin t x e t 2 2 n 0 1 n t 2 n 1 2 n 1 H 2 n 1 x displaystyle cos tx e frac t 2 2 sum n 0 infty 1 n frac t 2n 2n H 2n x sin tx e frac t 2 2 sum n 0 infty 1 n frac t 2n 1 2n 1 H 2n 1 x nbsp Povnota RedaguvatiFormula Kristoffelya Darbu dlya polinomiv Ermita maye viglyad i 0 n H i x H i y i 2 i 1 n 2 n 1 H n y H n 1 x H n x H n 1 y x y displaystyle sum i 0 n frac H i x H i y i 2 i frac 1 n 2 n 1 frac H n y H n 1 x H n x H n 1 y x y nbsp Bilsh togo nastupna formula spravdzhuyetsya i dlya uzagalnenij funkcij 2 n 0 ps n x ps n y d x y displaystyle sum n 0 infty psi n x psi n y delta x y nbsp de d delta funkciya Diraka psn funkciyi Ermita Cya uzagalnena formula sliduye yaksho poklasti u 1 u formuli Melera dijsnij pri 1 lt u lt 1 E x y u n 0 u n ps n x ps n y 1 p 1 u 2 e x p 1 u 1 u x y 2 4 1 u 1 u x y 2 4 displaystyle E x y u sum n 0 infty u n psi n x psi n y frac 1 sqrt pi 1 u 2 mathrm exp left frac 1 u 1 u frac x y 2 4 frac 1 u 1 u frac x y 2 4 right nbsp yaku mozhna ekvivalentno zapisati tak n 0 H n x H n y n u 2 n 1 1 u 2 e 2 u 1 u x y u 2 1 u 2 x y 2 displaystyle sum n 0 infty frac H n x H n y n left frac u 2 right n frac 1 sqrt 1 u 2 mathrm e frac 2u 1 u xy frac u 2 1 u 2 x y 2 nbsp Funkciya x y E x y u ye gustinoyu dlya miri Gausa na R2 yaka ye koli u pryamuye do 1 duzhe skoncentrovanoyu bilya liniyi y x i silno spadaye poza neyu Tomu n 0 u n f ps n ps n g E x y u f x g y d x d y f x g x d x f g displaystyle left langle left sum n 0 infty u n langle f psi n rangle psi n right g right rangle int int E x y u f x overline g y mathrm d x mathrm d y rightarrow int f x overline g x mathrm d x langle f g rangle nbsp koli ƒ g ye neperervnimi funkciyami na kompaktnomu nosiyi Ce privodit do togo sho ƒ mozhe buti virazhena cherez funkciyi Ermita u viglyadi sumi ryadu vektoriv z L2 R tobto f n 0 f ps n ps n displaystyle f sum n 0 infty langle f psi n rangle psi n nbsp Shob dovesti vishenavedenu rivnist dlya E x y u treba dekilka raziv vikoristati Fur ye peretvorennya funkciyi Gausa r p e r 2 x 2 4 e i s x s 2 r 2 d s r gt 0 displaystyle rho sqrt pi mathrm e rho 2 x 2 4 int mathrm e isx s 2 rho 2 mathrm d s quad rho gt 0 nbsp Polinomi Ermita mozhut buti predstavlennya u viglyadi H n x 1 n e x 2 d n d x n 1 2 p e i s x s 2 4 d s 1 n e x 2 1 2 p i s n e i s x s 2 4 d s displaystyle H n x 1 n mathrm e x 2 frac mathrm d n mathrm d x n Bigl frac 1 2 sqrt pi int mathrm e isx s 2 4 mathrm d s Bigr 1 n mathrm e x 2 frac 1 2 sqrt pi int is n mathrm e isx s 2 4 mathrm d s nbsp Z cim predstavlennyam dlya Hn x i Hn y mozhna bachiti sho E x y u n 0 u n 2 n n p H n x H n y e x 2 y 2 2 e x 2 y 2 2 4 p p n 0 1 2 n n u s t n e i s x i t y s 2 4 t 2 4 d s d t e x 2 y 2 2 4 p p e u s t 2 e i s x i t y s 2 4 t 2 4 d s d t displaystyle begin aligned E x y u amp sum n 0 infty frac u n 2 n n sqrt pi H n x H n y mathrm e x 2 y 2 2 amp frac mathrm e x 2 y 2 2 4 pi sqrt pi int int Bigl sum n 0 infty frac 1 2 n n ust n Bigr mathrm e isx ity s 2 4 t 2 4 mathrm d s mathrm d t amp frac mathrm e x 2 y 2 2 4 pi sqrt pi int int mathrm e ust 2 mathrm e isx ity s 2 4 t 2 4 mathrm d s mathrm d t end aligned nbsp a ce privodit do potribnogo rezultatu yaksho skoristatisya formuloyu peretvorennya Fur ye Gausovogo yadra pislya vikonannya pidstanovki s s t 2 t s t 2 displaystyle s frac sigma tau sqrt 2 qquad qquad t frac sigma tau sqrt 2 nbsp Diferencialni rivnyannya RedaguvatiPolinomi Ermita H n x displaystyle H n x nbsp ye rozv yazkami linijnogo diferencialnogo rivnyannya y x x y x n y x 0 displaystyle y x xy x ny x 0 nbsp Yaksho n displaystyle n nbsp ye cilim chislom to zagalnij rozv yazok vishenavedenogo rivnyannya zapisuyetsya yaky x A H n x B h n x displaystyle y x AH n x Bh n x nbsp de A B displaystyle A B nbsp dovilni stali a funkciyi h n x displaystyle h n x nbsp nazivayutsya funkciyami Ermita drugogo rodu Ci funkciyi ne zvodyatsya do polinomiv i yih mozhna viraziti lishe za dopomogoyu transcendentnih funkcij e x 2 2 displaystyle e x 2 2 nbsp ta 0 x e z 2 2 d z displaystyle int 0 x e z 2 2 dz nbsp Predstavlennya RedaguvatiPolinomi Ermita dopuskayut taki predstavlennya H n x n 2 p i G e z x z 2 2 z n 1 d z displaystyle H n x frac n 2 pi i oint Gamma frac e zx z 2 2 z n 1 dz nbsp de G displaystyle Gamma nbsp kontur sho ohoplyuye pochatok koordinat Inshe predstavlennya maye viglyad H n x 1 2 p x i y n e y 2 2 d y displaystyle H n x frac 1 sqrt 2 pi int infty infty x iy n e frac y 2 2 dy nbsp Zv yazok z inshimi specialnimi funkciyami RedaguvatiZv yazok z funkciyeyu Kummera H 2 n x 1 n 2 n 2 n n 1 F 1 n 1 2 x 2 2 H 2 n 1 x 1 n 2 n 2 n 1 n x 1 F 1 n 3 2 x 2 2 displaystyle H 2n x frac 1 n 2 n frac 2n n 1 F 1 left n frac 1 2 frac x 2 2 right H 2n 1 x frac 1 n 2 n frac 2n 1 n x 1 F 1 left n frac 3 2 frac x 2 2 right nbsp Zv yazok z polinomami Lagerra H 2 n x 2 n n L n 1 2 x 2 2 H 2 n 1 x 2 n n x L n 1 2 x 2 2 displaystyle H 2n x 2 n n L n 1 2 x 2 2 H 2n 1 x 2 n n x L n 1 2 x 2 2 nbsp Tvirna funkciya polinomiv Ermita maye viglyad g x t exp t 2 2 t x exp x 2 exp t x 2 displaystyle g x t exp t 2 2tx exp x 2 exp t x 2 nbsp Dlya ciyeyi funkciyi exp x 2 exp 1 x 2 n 0 H n x n t n displaystyle exp x 2 exp 1 x 2 sum n 0 infty frac H n x n t n nbsp Diferenciyuvannya n 0 H n x n t n displaystyle sum n 0 infty frac H n x n t n nbsp raziv n displaystyle n nbsp po t displaystyle t nbsp dlya livoyi chastini daye exp x 2 n t n exp t x 2 exp x 2 1 n n x n exp t x 2 displaystyle exp x 2 frac partial n partial t n exp t x 2 exp x 2 1 n frac partial n partial x n exp t x 2 nbsp a pravoruch H n x H n 1 x t H n 2 x t 2 displaystyle H n x H n 1 x t H n 2 x t 2 nbsp Vvazhayuchi t 0 displaystyle t 0 nbsp H n x 1 n exp x 2 d n d x n exp x 2 displaystyle H n x 1 n exp x 2 frac d n dx n exp x 2 nbsp oskilki t exp t x 2 x exp t x 2 displaystyle frac partial partial t exp t x 2 frac partial partial x exp t x 2 nbsp Takim chinom n displaystyle n nbsp diferenciyuvannya po x displaystyle x nbsp eksponencijnoyi funkciyi exp x 2 displaystyle exp x 2 nbsp privodit do polinomiv Ermita H n x displaystyle H n x nbsp Rekurentne spivvidnoshennya Prodiferenciyujmo g x t exp t 2 2 t x exp x 2 exp t x 2 displaystyle g x t exp t 2 2tx exp x 2 exp t x 2 nbsp po t displaystyle t nbsp g i 2 t x g x t displaystyle frac partial g partial i 2 t x g x t nbsp n 1 H n x n 1 t n 1 2 t x n 0 H n x n 0 displaystyle sum n 1 infty frac H n x n 1 t n 1 2 t x sum n 0 infty frac H n x n 0 nbsp n 0 H n 1 x n 2 x H n x n 2 H n 1 x n 1 t n 0 displaystyle sum n 0 infty frac H n 1 x n 2x frac H n x n frac 2H n 1 x n 1 t n 0 nbsp ta otrimajmoH n 1 x 2 x H n x 2 n H n 1 x 0 displaystyle H n 1 x 2xH n x 2nH n 1 x 0 nbsp Iz skazanogo mozhna otrimati diferencialne rivnyannyaH n x 2 x H n x 2 n H n 0 n 0 1 2 displaystyle H n prime prime x 2xH n x 2nH n 0 quad quad n overline 0 1 2 nbsp yake ye chastkovim rishennyam linijnogo diferencialnogo rivnyannya drugogo poryadku H n x 2 x H n x 2 n H n x 0 displaystyle H n prime prime x 2xH n prime x 2nH n x 0 nbsp Zastosuvannya RedaguvatiV kvantovij mehanici polinomi Ermita vhodyat do virazu hvilovoyi funkciyi kvantovogo garmonichnogo oscilyatora V bezrozmirnih zminnih rivnyannya Shredingera yake opisuye stani kvantovogo garmonichnogo oscilyatora maye viglyad d 2 d x 2 x 2 ps n x l n ps n x displaystyle left frac d 2 dx 2 x 2 right psi n x lambda n psi n x nbsp Rozv yazkami cogo rivnyannya ye vlasni funkciyi oscilyatora sho vidpovidayut vlasnim znachennyam l n 2 n 1 displaystyle lambda n 2n 1 nbsp Normovani na odinicyu voni zapisuyutsya yak ps n x e x 2 2 1 n 2 n n p H n x n 0 1 2 displaystyle psi n x e frac x 2 2 frac 1 n sqrt 2 n n sqrt pi H n x n 0 1 2 dots nbsp Zaznachimo sho v danomu virazi vikoristovuyutsya same fizichni polinomi Ermita H n x displaystyle H n x nbsp Polinomi Ermita vikoristovuyutsya v rozv yazku odnovimirnogo rivnyannya teploprovidnosti u t u x x 0 displaystyle u t u xx 0 nbsp na neskinchennomu intervali Ce rivnyannya maye rozv yazok u viglyadi eksponencijnoyi funkciyi u x t e a x a 2 t displaystyle u x t e alpha x alpha 2 t nbsp Oskilki taku funkciyu mozhna predstaviti u viglyadi rozkladu po polinomah Ermita a z inshogo boku vona mozhe buti rozkladena v ryad Tejlora po a displaystyle alpha nbsp e a x a 2 t n 0 a n n P n x t displaystyle e alpha x alpha 2 t sum n 0 infty frac alpha n n P n x t nbsp to funkciyi P n x t displaystyle P n x t nbsp sho ye rozv yazkami rivnyannya teploprovidnosti i zadovolnyayut pochatkovij umovi P n x t 0 x n displaystyle P n x t 0 x n nbsp virazhayutsya cherez polinomi Ermita nastupnim chinom P n x t i 2 t n H n x i 2 t 1 4 p t e x y 2 4 t y n d y displaystyle P n x t i sqrt 2t n H n left frac x i sqrt 2t right frac 1 sqrt 4 pi t int infty infty e frac x y 2 4t y n dy nbsp Dlya otrimannya ostannoyi rivnosti bulo vikoristano integral Puasona Fur ye V teoriyi jmovirnostej polinomi Ermita vhodyat do tak zvanih ryadiv Edzhvorta yaki vikoristovuyutsya dlya nablizhennya funkciyi gustini jmovirnosti cherez yiyi kumulyanti Primitki Redaguvati Hermite C Sur un nouveau developpement en serie de fonctions Compt Rend Acad Sci Paris 1864 T 58 S 93 100 266 273 peredrukovano takozh v C Hermite 1908 Oeuvres completes francuzka tome 2 Paris s 293 308 Wiener 1958 Literatura RedaguvatiAbramowitz Milton amp Stegun Irene A eds 1965 Chapter 22 Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables New York Dover ISBN 0 486 61272 4 Wiener Norbert 1958 The Fourier Integral and Certain of its Applications New York Dover Publications ISBN 0 486 60272 9 Whittaker E T Watson G N 1962 A Course of Modern Analysis London Cambridge University Press Zh Kampe de Fere R Kempbell G Peto T Fogel 1963 IX Funkcii matematicheskoj fiziki rosijska Moskva Fizmatgiz s 62 70 Zovnishni posilannya RedaguvatiEric W Weisstein Hermite Polynomial Arhivovano 21 lipnya 2008 u Wayback Machine angl na sajti MathWorld Arhivovano 28 chervnya 2017 u Wayback Machine Module for Hermite Polynomial Interpolation by John H Mathews nbsp Cya stattya nalezhit do dobrih statej ukrayinskoyi Vikipediyi Otrimano z https uk wikipedia org w index php title Polinomi Ermita amp oldid 39421188
