§10.12 Generating Function and Associated Series

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Keywords:: Bessel functions, Jacobi–Anger expansions, generating functions
Notes:: For (10.12.1) see Olver (1997b, pp. 55–56). For (10.12.2)–(10.12.6) set $t=e^{i\theta}$ and $ie^{i\theta}$ , and apply other straighforward substitutions, including differentiations with respect to $\theta$ in the case of (10.12.6). See also Watson (1944, pp. 22–23).
Referenced by:: §10.23(ii), §10.23(ii), Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/10.12
Addition (effective with 1.0.7):: Equations (10.12.2) and (10.12.3) have been identified as Jacobi–Anger expansions.
Suggested 2012-12-28 by Alexander Barnett
See also:: Annotations for Ch.10

For $z\in\mathbb{C}$ and $t\in\mathbb{C}\setminus\{0\}$ ,

10.12.1 $e^{\frac{1}{2}z(t-t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}J_{m}\left(z\right).$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, : integer and : complex variable
A&S Ref:: 9.1.41
Referenced by:: §10.12, §10.23(iii), §10.35
Permalink:: http://dlmf.nist.gov/10.12.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.12 and Ch.10

Jacobi–Anger expansions: for $z,\theta\in\mathbb{C}$ ,

10.12.2

$\cos\left(z\sin\theta\right)=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}J_{2k}% \left(z\right)\cos\left(2k\theta\right),$

$\sin\left(z\sin\theta\right)=2\sum_{k=0}^{\infty}J_{2k+1}\left(z\right)\sin% \left((2k+1)\theta\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : nonnegative integer and : complex variable
A&S Ref:: 9.1.42, 9.1.43
Referenced by:: §10.12
Permalink:: http://dlmf.nist.gov/10.12.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.12 and Ch.10

10.12.3

$\cos\left(z\cos\theta\right)=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}(-1)^{k}J% _{2k}\left(z\right)\cos\left(2k\theta\right),$

$\sin\left(z\cos\theta\right)=2\sum_{k=0}^{\infty}(-1)^{k}J_{2k+1}\left(z\right% )\cos\left((2k+1)\theta\right).$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : nonnegative integer and : complex variable
A&S Ref:: 9.1.44, 9.1.45
Referenced by:: §10.12
Permalink:: http://dlmf.nist.gov/10.12.E3
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.12 and Ch.10

10.12.4 $1=J_{0}\left(z\right)+2J_{2}\left(z\right)+2J_{4}\left(z\right)+2J_{6}\left(z% \right)+\dotsb,$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind and : complex variable
A&S Ref:: 9.1.46
Referenced by:: §10.74(iv)
Permalink:: http://dlmf.nist.gov/10.12.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.12 and Ch.10

10.12.5

$\cos z=J_{0}\left(z\right)-2J_{2}\left(z\right)+2J_{4}\left(z\right)-2J_{6}% \left(z\right)+\dotsb,$

$\sin z=2J_{1}\left(z\right)-2J_{3}\left(z\right)+2J_{5}\left(z\right)-\dotsb,$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and : complex variable
A&S Ref:: 9.1.47, 9.1.48
Permalink:: http://dlmf.nist.gov/10.12.E5
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.12 and Ch.10

10.12.6

$\tfrac{1}{2}z\cos z=J_{1}\left(z\right)-9J_{3}\left(z\right)+25J_{5}\left(z% \right)-49J_{7}\left(z\right)+\dotsb,$

$\tfrac{1}{2}z\sin z=4J_{2}\left(z\right)-16J_{4}\left(z\right)+36J_{6}\left(z% \right)-\dotsi.$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and : complex variable
Referenced by:: §10.12, §10.23(iii)
Permalink:: http://dlmf.nist.gov/10.12.E6
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.12 and Ch.10