# §28.11 Expansions in Series of Mathieu Functions

Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-infinite strip $S$ that contains the real axis, and $q$ be a normal value (§28.7). Then

 28.11.1 $f(z)=\alpha_{0}\operatorname{ce}_{0}\left(z,q\right)+\sum_{n=1}^{\infty}\left(% \alpha_{n}\operatorname{ce}_{n}\left(z,q\right)+\beta_{n}\operatorname{se}_{n}% \left(z,q\right)\right),$ ⓘ Defines: $f(z)$: function (locally) Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $q=h^{2}$: parameter, $n$: integer, $z$: complex variable, $\alpha_{n}$ and $\beta_{n}$ Referenced by: §28.11 Permalink: http://dlmf.nist.gov/28.11.E1 Encodings: TeX, pMML, png See also: Annotations for §28.11 and Ch.28

where

 28.11.2 $\displaystyle\alpha_{n}$ $\displaystyle=\frac{1}{\pi}\int_{0}^{2\pi}f(x)\operatorname{ce}_{n}\left(x,q% \right)\,\mathrm{d}x,$ $\displaystyle\beta_{n}$ $\displaystyle=\frac{1}{\pi}\int_{0}^{2\pi}f(x)\operatorname{se}_{n}\left(x,q% \right)\,\mathrm{d}x.$ ⓘ Defines: $\alpha_{n}$ (locally) and $\beta_{n}$ (locally) Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $q=h^{2}$: parameter, $n$: integer, $x$: real variable and $f(z)$: function Permalink: http://dlmf.nist.gov/28.11.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.11 and Ch.28

The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip $S$. See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of $q$ see Meixner et al. (1980, p. 33).

## Examples

With the notation of §28.4,

 28.11.3 $1=2\sum_{n=0}^{\infty}A_{0}^{2n}(q)\operatorname{ce}_{2n}\left(z,q\right),$
 28.11.4 $\cos 2mz=\sum_{n=0}^{\infty}A_{2m}^{2n}(q)\operatorname{ce}_{2n}\left(z,q% \right),$ $m\neq 0$,
 28.11.5 $\displaystyle\cos(2m+1)z$ $\displaystyle=\sum_{n=0}^{\infty}A_{2m+1}^{2n+1}(q)\operatorname{ce}_{2n+1}% \left(z,q\right),$ 28.11.6 $\displaystyle\sin(2m+1)z$ $\displaystyle=\sum_{n=0}^{\infty}B_{2m+1}^{2n+1}(q)\operatorname{se}_{2n+1}% \left(z,q\right),$ 28.11.7 $\displaystyle\sin(2m+2)z$ $\displaystyle=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)\operatorname{se}_{2n+2}% \left(z,q\right).$