In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case , are a sequence of polynomials in used to expand functions in term of Bessel functions.[1]
neumann, polynomial, mathematics, introduced, carl, neumann, special, case, displaystyle, alpha, sequence, polynomials, displaystyle, used, expand, functions, term, bessel, functions, first, polynomials, displaystyle, alpha, frac, displaystyle, alpha, frac, al. In mathematics the Neumann polynomials introduced by Carl Neumann for the special case a 0 displaystyle alpha 0 are a sequence of polynomials in 1 t displaystyle 1 t used to expand functions in term of Bessel functions 1 The first few polynomials are O 0 a t 1 t displaystyle O 0 alpha t frac 1 t O 1 a t 2 a 1 t 2 displaystyle O 1 alpha t 2 frac alpha 1 t 2 O 2 a t 2 a t 4 2 a 1 a t 3 displaystyle O 2 alpha t frac 2 alpha t 4 frac 2 alpha 1 alpha t 3 O 3 a t 2 1 a 3 a t 2 8 1 a 2 a 3 a t 4 displaystyle O 3 alpha t 2 frac 1 alpha 3 alpha t 2 8 frac 1 alpha 2 alpha 3 alpha t 4 O 4 a t 1 a 4 a 2 t 4 1 a 2 a 4 a t 3 16 1 a 2 a 3 a 4 a t 5 displaystyle O 4 alpha t frac 1 alpha 4 alpha 2t 4 frac 1 alpha 2 alpha 4 alpha t 3 16 frac 1 alpha 2 alpha 3 alpha 4 alpha t 5 A general form for the polynomial is O n a t a n 2 a k 0 n 2 1 n k n k k a n k 2 t n 1 2 k displaystyle O n alpha t frac alpha n 2 alpha sum k 0 lfloor n 2 rfloor 1 n k frac n k k alpha choose n k left frac 2 t right n 1 2k and they have the generating function z 2 a G a 1 1 t z n 0 O n a t J a n z displaystyle frac left frac z 2 right alpha Gamma alpha 1 frac 1 t z sum n 0 O n alpha t J alpha n z where J are Bessel functions To expand a function f in the form f z n 0 a n J a n z displaystyle f z sum n 0 a n J alpha n z for z lt c displaystyle z lt c compute a n 1 2 p i z c G a 1 z 2 a f z O n a z d z displaystyle a n frac 1 2 pi i oint z c frac Gamma alpha 1 left frac z 2 right alpha f z O n alpha z dz where c lt c displaystyle c lt c and c is the distance of the nearest singularity of z a f z displaystyle z alpha f z from z 0 displaystyle z 0 Examples editAn example is the extension 1 2 z s G s k 0 1 k J s 2 k z s 2 k s k displaystyle left tfrac 1 2 z right s Gamma s cdot sum k 0 1 k J s 2k z s 2k s choose k nbsp or the more general Sonine formula 2 e i g z G s k 0 i k C k s g s k J s k z z 2 s displaystyle e i gamma z Gamma s cdot sum k 0 i k C k s gamma s k frac J s k z left frac z 2 right s nbsp where C k s displaystyle C k s nbsp is Gegenbauer s polynomial Then citation needed original research z 2 2 k 2 k 1 J s z i k 1 i k i k 1 2 k 1 i k s 1 2 k 1 s 2 i J s 2 i z displaystyle frac left frac z 2 right 2k 2k 1 J s z sum i k 1 i k i k 1 choose 2k 1 i k s 1 choose 2k 1 s 2i J s 2i z nbsp n 0 t n J s n z e t z 2 t s j 0 z 2 t j j g j s t z 2 G j s 0 e z x 2 2 t z x t J s z 1 x 2 1 x 2 s d x displaystyle sum n 0 t n J s n z frac e frac tz 2 t s sum j 0 frac left frac z 2t right j j frac gamma left j s frac tz 2 right Gamma j s int 0 infty e frac zx 2 2t frac zx t frac J s z sqrt 1 x 2 sqrt 1 x 2 s dx nbsp the confluent hypergeometric function M a s z G s k 0 1 t k L k a k t J s k 1 2 t z t z s k 1 displaystyle M a s z Gamma s sum k 0 infty left frac 1 t right k L k a k t frac J s k 1 left 2 sqrt tz right sqrt tz s k 1 nbsp and in particular J s 2 z z s 4 s G s 1 2 p e 2 i z k 0 L k s 1 2 k i t 4 4 i z k J 2 s k 2 t z t z 2 s k displaystyle frac J s 2z z s frac 4 s Gamma left s frac 1 2 right sqrt pi e 2iz sum k 0 L k s 1 2 k left frac it 4 right 4iz k frac J 2s k left 2 sqrt tz right sqrt tz 2s k nbsp the index shift formula G n m J n z G m 1 n 0 G n m n n G n n 1 z 2 n m n J m n z displaystyle Gamma nu mu J nu z Gamma mu 1 sum n 0 frac Gamma nu mu n n Gamma nu n 1 left frac z 2 right nu mu n J mu n z nbsp the Taylor expansion addition formula J s z 2 2 u z z 2 2 u z s k 0 u k k J s k z z s displaystyle frac J s left sqrt z 2 2uz right left sqrt z 2 2uz right pm s sum k 0 frac pm u k k frac J s pm k z z pm s nbsp cf 3 failed verification and the expansion of the integral of the Bessel function J s z d z 2 k 0 J s 2 k 1 z displaystyle int J s z dz 2 sum k 0 J s 2k 1 z nbsp are of the same type See also editBessel function Bessel polynomial Lommel polynomial Hankel transform Fourier Bessel series Schlafli polynomialNotes edit Abramowitz and Stegun p 363 9 1 82 ff Erdelyi et al 1955harvnb error no target CITEREFErdelyiMagnusOberhettingerTricomi1955 help II 7 10 1 p 64 Gradshteyn Izrail Solomonovich Ryzhik Iosif Moiseevich Geronimus Yuri Veniaminovich Tseytlin Michail Yulyevich Jeffrey Alan 2015 October 2014 8 515 1 In Zwillinger Daniel Moll Victor Hugo eds Table of Integrals Series and Products Translated by Scripta Technica Inc 8 ed Academic Press Inc p 944 ISBN 0 12 384933 0 LCCN 2014010276 Retrieved from https en wikipedia org w index php title Neumann polynomial amp oldid 1069139071, wikipedia, wiki, book, books, library,