# §10.12 Generating Function and Associated Series

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

 10.12.1 $e^{\frac{1}{2}z(t-t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}\mathop{J_{m}\/}% \nolimits\!\left(z\right).$ Symbols: $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$: Bessel function of the first kind, $e$: base of exponential function, $m$: 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

Jacobi–Anger expansions: for $z,\theta\in\Complex$,

 10.12.2 $\displaystyle\mathop{\cos\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)$ $\displaystyle=\mathop{J_{0}\/}\nolimits\!\left(z\right)+2\sum_{k=1}^{\infty}% \mathop{J_{2k}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!\left(2k% \theta\right),$ $\displaystyle\mathop{\sin\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)$ $\displaystyle=2\sum_{k=0}^{\infty}\mathop{J_{2k+1}\/}\nolimits\!\left(z\right)% \mathop{\sin\/}\nolimits\!\left((2k+1)\theta\right),$
 10.12.3 $\displaystyle\mathop{\cos\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)$ $\displaystyle=\mathop{J_{0}\/}\nolimits\!\left(z\right)+2\sum_{k=1}^{\infty}(-% 1)^{k}\mathop{J_{2k}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!% \left(2k\theta\right),$ $\displaystyle\mathop{\sin\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)$ $\displaystyle=2\sum_{k=0}^{\infty}(-1)^{k}\mathop{J_{2k+1}\/}\nolimits\!\left(% z\right)\mathop{\cos\/}\nolimits\!\left((2k+1)\theta\right).$
 10.12.4 $1=\mathop{J_{0}\/}\nolimits\!\left(z\right)+2\!\mathop{J_{2}\/}\nolimits\!% \left(z\right)+2\!\mathop{J_{4}\/}\nolimits\!\left(z\right)+2\!\mathop{J_{6}\/% }\nolimits\!\left(z\right)+\cdots,$ Symbols: $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$: 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
 10.12.5 $\displaystyle\mathop{\cos\/}\nolimits z$ $\displaystyle=\mathop{J_{0}\/}\nolimits\!\left(z\right)-2\!\mathop{J_{2}\/}% \nolimits\!\left(z\right)+2\!\mathop{J_{4}\/}\nolimits\!\left(z\right)-2\!% \mathop{J_{6}\/}\nolimits\!\left(z\right)+\cdots,$ $\displaystyle\mathop{\sin\/}\nolimits z$ $\displaystyle=2\!\mathop{J_{1}\/}\nolimits\!\left(z\right)-2\!\mathop{J_{3}\/}% \nolimits\!\left(z\right)+2\!\mathop{J_{5}\/}\nolimits\!\left(z\right)-\cdots,$
 10.12.6 $\displaystyle\tfrac{1}{2}z\mathop{\cos\/}\nolimits z$ $\displaystyle=\mathop{J_{1}\/}\nolimits\!\left(z\right)-9\!\mathop{J_{3}\/}% \nolimits\!\left(z\right)+25\!\mathop{J_{5}\/}\nolimits\!\left(z\right)-49\!% \mathop{J_{7}\/}\nolimits\!\left(z\right)+\cdots,$ $\displaystyle\tfrac{1}{2}z\mathop{\sin\/}\nolimits z$ $\displaystyle=4\!\mathop{J_{2}\/}\nolimits\!\left(z\right)-16\!\mathop{J_{4}\/% }\nolimits\!\left(z\right)+36\!\mathop{J_{6}\/}\nolimits\!\left(z\right)-\cdots.$