# §10.12 Generating Function and Associated Series

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

 10.12.1 $e^{\frac{1}{2}z(t-t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}J_{m}\left(z\right).$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathrm{e}$: 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 See also: Annotations for 10.12 and 10

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

 10.12.2 $\displaystyle\cos\left(z\sin\theta\right)$ $\displaystyle=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}J_{2k}\left(z\right)\cos% \left(2k\theta\right),$ $\displaystyle\sin\left(z\sin\theta\right)$ $\displaystyle=2\sum_{k=0}^{\infty}J_{2k+1}\left(z\right)\sin\left((2k+1)\theta% \right),$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $k$: nonnegative integer and $z$: complex variable A&S Ref: 9.1.42, 9.1.43 Referenced by: §10.12 Permalink: http://dlmf.nist.gov/10.12.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.12 and 10
 10.12.3 $\displaystyle\cos\left(z\cos\theta\right)$ $\displaystyle=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}(-1)^{k}J_{2k}\left(z% \right)\cos\left(2k\theta\right),$ $\displaystyle\sin\left(z\cos\theta\right)$ $\displaystyle=2\sum_{k=0}^{\infty}(-1)^{k}J_{2k+1}\left(z\right)\cos\left((2k+% 1)\theta\right).$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $k$: nonnegative integer and $z$: complex variable A&S Ref: 9.1.44, 9.1.45 Referenced by: §10.12 Permalink: http://dlmf.nist.gov/10.12.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.12 and 10
 10.12.4 $1=J_{0}\left(z\right)+2\!J_{2}\left(z\right)+2\!J_{4}\left(z\right)+2\!J_{6}% \left(z\right)+\cdots,$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: 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 See also: Annotations for 10.12 and 10
 10.12.5 $\displaystyle\cos z$ $\displaystyle=J_{0}\left(z\right)-2\!J_{2}\left(z\right)+2\!J_{4}\left(z\right% )-2\!J_{6}\left(z\right)+\cdots,$ $\displaystyle\sin z$ $\displaystyle=2\!J_{1}\left(z\right)-2\!J_{3}\left(z\right)+2\!J_{5}\left(z% \right)-\cdots,$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 9.1.47, 9.1.48 Permalink: http://dlmf.nist.gov/10.12.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.12 and 10
 10.12.6 $\displaystyle\tfrac{1}{2}z\cos z$ $\displaystyle=J_{1}\left(z\right)-9\!J_{3}\left(z\right)+25\!J_{5}\left(z% \right)-49\!J_{7}\left(z\right)+\cdots,$ $\displaystyle\tfrac{1}{2}z\sin z$ $\displaystyle=4\!J_{2}\left(z\right)-16\!J_{4}\left(z\right)+36\!J_{6}\left(z% \right)-\cdots.$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $z$: complex variable Referenced by: §10.12, §10.23(iii) Permalink: http://dlmf.nist.gov/10.12.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.12 and 10