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6000 DLMF: §10.53 Power Series ‣ Spherical Bessel Functions ‣ Chapter 10 Bessel Functions

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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

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