# §28.14 Fourier Series

The Fourier series

 28.14.1 $\displaystyle\mathrm{me}_{\nu}\left(z,q\right)$ $\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)e^{\mathrm{i}(\nu+2m)z},$ ⓘ Symbols: $\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $q=h^{2}$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients A&S Ref: 20.3.8 (in slightly different form) Referenced by: §28.22(ii) Permalink: http://dlmf.nist.gov/28.14.E1 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28 28.14.2 $\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)$ $\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)\cos(\nu+2m)z,$ 28.14.3 $\displaystyle\mathrm{se}_{\nu}\left(z,q\right)$ $\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)\sin(\nu+2m)z,$

converge absolutely and uniformly on all compact sets in the $z$-plane. The coefficients satisfy

 28.14.4 ${qc_{2m+2}-\left(a-(\nu+2m)^{2}\right)c_{2m}+qc_{2m-2}=0},$ $a=\lambda_{\nu}\left(q\right),c_{2m}=c_{2m}^{\nu}(q)$,

and the normalization relation

 28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients Referenced by: item (d) Permalink: http://dlmf.nist.gov/28.14.E5 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28

compare (28.12.5). Ambiguities in sign are resolved by (28.14.9) when $q=0$, and by continuity for other values of $q$.

The rate of convergence is indicated by

 28.14.6 $\frac{c^{\nu}_{2m}(q)}{c^{\nu}_{2m\mp 2}(q)}=\frac{-q}{4m^{2}}\left(1+O\left(% \frac{1}{m}\right)\right),$ $m\to\pm\infty$.

For changes of sign of $\nu$, $q$, and $m$,

 28.14.7 $\displaystyle c_{-2m}^{-\nu}(q)$ $\displaystyle=c_{2m}^{\nu}(q),$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients Permalink: http://dlmf.nist.gov/28.14.E7 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28 28.14.8 $\displaystyle c_{2m}^{\nu}(-q)$ $\displaystyle=(-1)^{m}c_{2m}^{\nu}(q).$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients Permalink: http://dlmf.nist.gov/28.14.E8 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28

When $q=0$,

 28.14.9 $\displaystyle c_{0}^{\nu}(0)$ $\displaystyle=1,$ $\displaystyle c_{2m}^{\nu}(0)$ $\displaystyle=0,$ $m\neq 0$. ⓘ Symbols: $m$: integer, $\nu$: complex parameter and $c_{2m}(q)$: coefficients A&S Ref: 20.8.2 Referenced by: §28.14 Permalink: http://dlmf.nist.gov/28.14.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.14 and Ch.28

When $q\to 0$ with $m$ ($\geq 1$) and $\nu$ fixed,

 28.14.10 $c_{2m}^{\nu}(q)=\left(\frac{(-1)^{m}q^{m}\Gamma\left(\nu+1\right)}{m!\,2^{2m}% \Gamma\left(\nu+m+1\right)}+O\left(q^{m+2}\right)\right)c_{0}^{\nu}(q).$