§10.56 Generating Functions

Notes:: To verify (10.56.1) and (10.56.2) show that each side of both equations satisfies the differential equation $(2t-z)(\ifrac{{d}^{2}w}{{dt}^{2}})+(\ifrac{dw}{dt})=zw$ 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).
A&S Ref:: 10.1.39, 10.1.40 (modified) 10.2.30, 10.2.31
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.56

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

10.56.1 $\frac{\mathop{\cos\/}\nolimits\sqrt{z^{2}-2zt}}{z}=\frac{\mathop{\cos\/}\nolimits z}{z}+\sum _{{n=1}}^{\infty}\frac{t^{n}}{n!}\mathop{\mathsf{j}_{{n-1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $!$ : $n!$ : factorial, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : 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
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10.56.2 $\frac{\mathop{\sin\/}\nolimits\sqrt{z^{2}-2zt}}{z}=\frac{\mathop{\sin\/}\nolimits z}{z}+\sum _{{n=1}}^{\infty}\frac{t^{n}}{n!}\mathop{\mathsf{y}_{{n-1}}\/}\nolimits\!\left(z\right).$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : 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
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10.56.3 $\frac{\mathop{\cosh\/}\nolimits\sqrt{z^{2}+2izt}}{z}=\frac{\mathop{\cosh\/}\nolimits z}{z}+\sum _{{n=1}}^{\infty}\frac{(it)^{n}}{n!}\mathop{{\mathsf{i}^{{(1)}}_{{n-1}}}\/}\nolimits\!\left(z\right),$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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

10.56.4 $\frac{\mathop{\sinh\/}\nolimits\sqrt{z^{2}+2izt}}{z}=\frac{\mathop{\sinh\/}\nolimits z}{z}+\sum _{{n=1}}^{\infty}\frac{(it)^{n}}{n!}\mathop{{\mathsf{i}^{{(2)}}_{{n-1}}}\/}\nolimits\!\left(z\right),$

Symbols:: $!$ : $n!$ : factorial, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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

10.56.5 $\frac{\mathop{\exp\/}\nolimits\!\left(-\sqrt{z^{2}+2izt}\right)}{z}=\frac{e^{{-z}}}{z}+\frac{2}{\pi}\sum _{{n=1}}^{\infty}\frac{(-it)^{n}}{n!}\mathop{\mathsf{k}_{{n-1}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
Referenced by:: §10.56, Ch.10
Permalink:: http://dlmf.nist.gov/10.56.E5
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