§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		$e^{\frac{1}{2}z(t-t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}J_{m}\left(z\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathrm{e}$ : base of natural logarithm, $m$ : integer and $z$ : 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	$\displaystyle\cos\left(z\sin\theta\right)$	$\displaystyle=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}J_{2k}\left(z\right)\cos% \left(2k\theta\right),$
10.12.2	$\displaystyle\sin\left(z\sin\theta\right)$	$\displaystyle=2\sum_{k=0}^{\infty}J_{2k+1}\left(z\right)\sin\left((2k+1)\theta% \right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer and $z$ : 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	$\displaystyle\cos\left(z\cos\theta\right)$	$\displaystyle=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}(-1)^{k}J_{2k}\left(z% \right)\cos\left(2k\theta\right),$
10.12.3	$\displaystyle\sin\left(z\cos\theta\right)$	$\displaystyle=2\sum_{k=0}^{\infty}(-1)^{k}J_{2k+1}\left(z\right)\cos\left((2k+% 1)\theta\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer and $z$ : 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		$1=J_{0}\left(z\right)+2J_{2}\left(z\right)+2J_{4}\left(z\right)+2J_{6}\left(z% \right)+\dotsb,$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind and $z$ : 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	$\displaystyle\cos z$	$\displaystyle=J_{0}\left(z\right)-2J_{2}\left(z\right)+2J_{4}\left(z\right)-2J% _{6}\left(z\right)+\dotsb,$
10.12.5	$\displaystyle\sin z$	$\displaystyle=2J_{1}\left(z\right)-2J_{3}\left(z\right)+2J_{5}\left(z\right)-\dotsb,$
ⓘ 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 $z$ : 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	$\displaystyle\tfrac{1}{2}z\cos z$	$\displaystyle=J_{1}\left(z\right)-9J_{3}\left(z\right)+25J_{5}\left(z\right)-4% 9J_{7}\left(z\right)+\dotsb,$
10.12.6	$\displaystyle\tfrac{1}{2}z\sin z$	$\displaystyle=4J_{2}\left(z\right)-16J_{4}\left(z\right)+36J_{6}\left(z\right)% -\dotsi.$
ⓘ 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 $z$ : 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