# §28.19 Expansions in Series of $\mathop{\mathrm{me}_{\nu+2n}\/}\nolimits$ Functions

Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. Assume also

 28.19.1 $f(z+\pi)=e^{\mathrm{i}\nu\pi}f(z).$

Then

 28.19.2 $f(z)=\sum_{n=-\infty}^{\infty}f_{n}\mathop{\mathrm{me}_{\nu+2n}\/}\nolimits\!% \left(z,q\right),$ Defines: $f(z)$: function (locally) Symbols: $\mathop{\mathrm{me}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $q=h^{2}$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f_{n}$: coefficients Referenced by: §28.19 Permalink: http://dlmf.nist.gov/28.19.E2 Encodings: TeX, pMML, png See also: Annotations for 28.19

where

 28.19.3 $f_{n}=\frac{1}{\pi}\int_{0}^{\pi}f(z)\mathop{\mathrm{me}_{\nu+2n}\/}\nolimits% \!\left(-z,q\right)\mathrm{d}z.$

The series (28.19.2) converges absolutely and uniformly on compact subsets within $S$.

## Example

 28.19.4 $e^{\mathrm{i}\nu z}=\sum_{n=-\infty}^{\infty}c^{\nu+2n}_{-2n}(q)\mathop{% \mathrm{me}_{\nu+2n}\/}\nolimits\!\left(z,q\right),$

where the coefficients are as in §28.14.