§10.35 Generating Function and Associated Series

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 exponential function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: 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 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 exponential function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: 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 10
 10.35.3 $e^{z\sin\theta}=I_{0}\left(z\right)+2\sum_{k=0}^{\infty}(-1)^{k}I_{2k+1}\left(% z\right)\sin\left((2k+1)\theta\right)+2\sum_{k=1}^{\infty}(-1)^{k}I_{2k}\left(% z\right)\cos\left(2k\theta\right).$
 10.35.4 $1=I_{0}\left(z\right)-2\!I_{2}\left(z\right)+2\!I_{4}\left(z\right)-2\!I_{6}% \left(z\right)+\cdots,$ ⓘ 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 10
 10.35.5 $e^{\pm z}=I_{0}\left(z\right)\pm 2\!I_{1}\left(z\right)+2\!I_{2}\left(z\right)% \pm 2\!I_{3}\left(z\right)+\cdots,$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: 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 10
 10.35.6 $\displaystyle\cosh z$ $\displaystyle=I_{0}\left(z\right)+2\!I_{2}\left(z\right)+2\!I_{4}\left(z\right% )+2\!I_{6}\left(z\right)+\dots,$ $\displaystyle\sinh z$ $\displaystyle=2\!I_{1}\left(z\right)+2\!I_{3}\left(z\right)+2\!I_{5}\left(z% \right)+\dots.$ ⓘ 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 10