§10.35 Generating Function and Associated Series

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Keywords:: Bessel Functions, Jacobi–Anger expansions, generating function, modified Bessel functions
Notes:: For (10.35.1) replace $z$ and $t$ in (10.12.1) by $i z$ and $-it$ , respectively, and apply (10.27.6). (10.35.2)–(10.35.6) are obtained by setting $t=e^{i\theta}$ , $t=-ie^{i\theta}$ , together with other straightforward substitutions.
Permalink:: http://dlmf.nist.gov/10.35
Addition (effective with 1.0.7):: Equations (10.35.2) and (10.35.3) have been identified as Jacobi-Anger expansions. Corresponding entries have been made in the index.
Suggested 2012-12-28 by Alexander Barnett
See also:: Annotations for Ch.10

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

10.35.1		$e^{\frac{1}{2}z(t+t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}I_{m}\left(z\right).$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $m$ : integer and $z$ : complex variable A&S Ref: 9.6.33 Referenced by: §10.35 Permalink: http://dlmf.nist.gov/10.35.E1 Encodings: TeX, pMML, png See also: Annotations for §10.35 and Ch.10

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

10.35.2		$e^{z\cos\theta}=I_{0}\left(z\right)+2\sum_{k=1}^{\infty}I_{k}\left(z\right)% \cos\left(k\theta\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.6.34 Referenced by: §10.35 Permalink: http://dlmf.nist.gov/10.35.E2 Encodings: TeX, pMML, png See also: Annotations for §10.35 and Ch.10

10.35.3		$e^{z\sin\theta}=I_{0}\left(z\right)+2\sum_{k=0}^{\infty}(-1)^{k}I_{2k+1}\left(% z\right)\sin\left((2k+1)\theta\right)+2\sum_{k=1}^{\infty}(-1)^{k}I_{2k}\left(% z\right)\cos\left(2k\theta\right).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.6.35 Referenced by: §10.35 Permalink: http://dlmf.nist.gov/10.35.E3 Encodings: TeX, pMML, png See also: Annotations for §10.35 and Ch.10

10.35.4		$1=I_{0}\left(z\right)-2I_{2}\left(z\right)+2I_{4}\left(z\right)-2I_{6}\left(z% \right)+\dotsb,$
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and $z$ : complex variable A&S Ref: 9.6.36 Permalink: http://dlmf.nist.gov/10.35.E4 Encodings: TeX, pMML, png See also: Annotations for §10.35 and Ch.10

10.35.5		$e^{\pm z}=I_{0}\left(z\right)\pm 2I_{1}\left(z\right)+2I_{2}\left(z\right)\pm 2% I_{3}\left(z\right)+\dotsb,$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and $z$ : complex variable A&S Ref: 9.6.37,9.6.38 Permalink: http://dlmf.nist.gov/10.35.E5 Encodings: TeX, pMML, png See also: Annotations for §10.35 and Ch.10

10.35.6	$\displaystyle\cosh z$	$\displaystyle=I_{0}\left(z\right)+2I_{2}\left(z\right)+2I_{4}\left(z\right)+2I% _{6}\left(z\right)+\dots,$
10.35.6	$\displaystyle\sinh z$	$\displaystyle=2I_{1}\left(z\right)+2I_{3}\left(z\right)+2I_{5}\left(z\right)+\dots.$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind and $z$ : complex variable A&S Ref: 9.6.39, 9.6.40 Referenced by: §10.35 Permalink: http://dlmf.nist.gov/10.35.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.35 and Ch.10