§10.56 Generating Functions

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Keywords:: generating functions, spherical Bessel functions
A&S Ref:: 10.1.39, 10.1.40 (modified) 10.2.30, 10.2.31
Notes:: To verify (10.56.1) and (10.56.2) show that each side of both equations satisfies the differential equation $(2t-z)(\ifrac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}})+(\ifrac{\mathrm{d}w}{% \mathrm{d}t})=zw$ via the first of (10.51.1) and (10.49.3), (10.49.5). Then check the initial conditions at $t=0$ . (10.56.3) and (10.56.4) follow from (10.56.1) and (10.56.2) via (10.47.12); then (10.56.5) follows from (10.47.11).
Permalink:: http://dlmf.nist.gov/10.56
See also:: Annotations for Ch.10

When $2|t|<|z|$ ,

10.56.1	$\displaystyle\frac{\cos\sqrt{z^{2}-2zt}}{z}$	$\displaystyle=\frac{\cos z}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\mathsf{j}_{% n-1}\left(z\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable Referenced by: §10.56, Ch.10 Permalink: http://dlmf.nist.gov/10.56.E1 Encodings: TeX, pMML, png See also: Annotations for §10.56 and Ch.10
10.56.2	$\displaystyle\frac{\sin\sqrt{z^{2}-2zt}}{z}$	$\displaystyle=\frac{\sin z}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\mathsf{y}_{% n-1}\left(z\right).$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer and $z$ : complex variable Referenced by: §10.56 Permalink: http://dlmf.nist.gov/10.56.E2 Encodings: TeX, pMML, png See also: Annotations for §10.56 and Ch.10

10.56.3	$\displaystyle\frac{\cosh\sqrt{z^{2}+2izt}}{z}$	$\displaystyle=\frac{\cosh z}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}{\mathsf% {i}^{(1)}_{n-1}}\left(z\right),$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable Referenced by: §10.56 Permalink: http://dlmf.nist.gov/10.56.E3 Encodings: TeX, pMML, png See also: Annotations for §10.56 and Ch.10
10.56.4	$\displaystyle\frac{\sinh\sqrt{z^{2}+2izt}}{z}$	$\displaystyle=\frac{\sinh z}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}{\mathsf% {i}^{(2)}_{n-1}}\left(z\right),$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable Referenced by: §10.56 Permalink: http://dlmf.nist.gov/10.56.E4 Encodings: TeX, pMML, png See also: Annotations for §10.56 and Ch.10

10.56.5		$\frac{\exp\left(-\sqrt{z^{2}+2izt}\right)}{z}=\frac{e^{-z}}{z}+\frac{2}{\pi}% \sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\mathsf{k}_{n-1}\left(z\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable Referenced by: §10.56, Ch.10 Permalink: http://dlmf.nist.gov/10.56.E5 Encodings: TeX, pMML, png See also: Annotations for §10.56 and Ch.10