10 Bessel Functions Spherical Bessel Functions 10.52 Limiting Forms 10.54 Integral Representations

§10.53 Power Series

ⓘ

Keywords:: power series, spherical Bessel functions
Notes:: Combine (10.2.2) and (10.25.2) with (10.47.3), (10.47.4), and (10.47.7).
Referenced by:: §10.52(i)
Permalink:: http://dlmf.nist.gov/10.53
See also:: Annotations for Ch.10

10.53.1		$\mathsf{j}_{n}\left(z\right)=z^{n}\sum_{k=0}^{\infty}\frac{(-\frac{1}{2}z^{2})% ^{k}}{k!(2n+2k+1)!!},$
ⓘ Symbols: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 10.1.2 Referenced by: §10.53 Permalink: http://dlmf.nist.gov/10.53.E1 Encodings: TeX, pMML, png See also: Annotations for §10.53 and Ch.10

10.53.2		$\mathsf{y}_{n}\left(z\right)=-\frac{1}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!% (\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}\sum_{k=n+1}^{\infty}% \frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.$
ⓘ Symbols: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 10.1.3 Referenced by: §10.53 Permalink: http://dlmf.nist.gov/10.53.E2 Encodings: TeX, pMML, png See also: Annotations for §10.53 and Ch.10

10.53.3	$\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)$	$\displaystyle=z^{n}\sum_{k=0}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1% )!!},$
ⓘ Symbols: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 10.2.5 Referenced by: §10.53 Permalink: http://dlmf.nist.gov/10.53.E3 Encodings: TeX, pMML, png See also: Annotations for §10.53 and Ch.10
10.53.4	$\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)$	$\displaystyle=\frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(-\frac{1% }{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(\frac{1}{2}z^% {2})^{k}}{k!(2k-2n-1)!!}.$
ⓘ Symbols: $!!$ : double factorial, $!$ : factorial (as in $n!$ ), ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 10.2.6 Referenced by: §10.53 Permalink: http://dlmf.nist.gov/10.53.E4 Encodings: TeX, pMML, png See also: Annotations for §10.53 and Ch.10

For ${\mathsf{h}^{(1)}_{n}}\left(z\right)$ and ${\mathsf{h}^{(2)}_{n}}\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). For $\mathsf{k}_{n}\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).

10.52 Limiting Forms 10.54 Integral Representations

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§10.53 Power Series

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.