10 Bessel Functions Modified Bessel Functions 10.34 Analytic Continuation 10.36 Other Differential Equations

§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 and in (10.12.1) by and , 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 $e^{\frac{1}{2}z(t+t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}I_{m}\left(z\right).$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : integer and : 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 $e^{z\cos\theta}=I_{0}\left(z\right)+2\sum_{k=1}^{\infty}I_{k}\left(z\right)% \cos\left(k\theta\right),$

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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, : nonnegative integer and : 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 $e^{z\sin\theta}=I_{0}\left(z\right)+2\sum_{k=0}^{\infty}(-1)^{k}I_{2k+1}\left(% z\right)\sin\left((2k+1)\theta\right)+2\sum_{k=1}^{\infty}(-1)^{k}I_{2k}\left(% z\right)\cos\left(2k\theta\right).$

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