# §10.35 Generating Function and Associated Series

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

 10.35.1 $e^{\frac{1}{2}z(t+t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}\mathop{I_{m}\/}% \nolimits\!\left(z\right).$ Symbols: $\mathrm{e}$: base of exponential function, $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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

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

 10.35.2 $e^{z\mathop{\cos\/}\nolimits\theta}=\mathop{I_{0}\/}\nolimits\!\left(z\right)+% 2\sum_{k=1}^{\infty}\mathop{I_{k}\/}\nolimits\!\left(z\right)\mathop{\cos\/}% \nolimits\!\left(k\theta\right),$
 10.35.3 $e^{z\mathop{\sin\/}\nolimits\theta}=\mathop{I_{0}\/}\nolimits\!\left(z\right)+% 2\sum_{k=0}^{\infty}(-1)^{k}\mathop{I_{2k+1}\/}\nolimits\!\left(z\right)% \mathop{\sin\/}\nolimits\!\left((2k+1)\theta\right)+2\sum_{k=1}^{\infty}(-1)^{% k}\mathop{I_{2k}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!\left(2k% \theta\right).$
 10.35.4 $1=\mathop{I_{0}\/}\nolimits\!\left(z\right)-2\!\mathop{I_{2}\/}\nolimits\!% \left(z\right)+2\!\mathop{I_{4}\/}\nolimits\!\left(z\right)-2\!\mathop{I_{6}\/% }\nolimits\!\left(z\right)+\cdots,$ Symbols: $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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
 10.35.5 $e^{\pm z}=\mathop{I_{0}\/}\nolimits\!\left(z\right)\pm 2\!\mathop{I_{1}\/}% \nolimits\!\left(z\right)+2\!\mathop{I_{2}\/}\nolimits\!\left(z\right)\pm 2\!% \mathop{I_{3}\/}\nolimits\!\left(z\right)+\cdots,$ Symbols: $\mathrm{e}$: base of exponential function, $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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
 10.35.6 $\displaystyle\mathop{\cosh\/}\nolimits z$ $\displaystyle=\mathop{I_{0}\/}\nolimits\!\left(z\right)+2\!\mathop{I_{2}\/}% \nolimits\!\left(z\right)+2\!\mathop{I_{4}\/}\nolimits\!\left(z\right)+2\!% \mathop{I_{6}\/}\nolimits\!\left(z\right)+\dots,$ $\displaystyle\mathop{\sinh\/}\nolimits z$ $\displaystyle=2\!\mathop{I_{1}\/}\nolimits\!\left(z\right)+2\!\mathop{I_{3}\/}% \nolimits\!\left(z\right)+2\!\mathop{I_{5}\/}\nolimits\!\left(z\right)+\dots.$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function, $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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