§28.11 Expansions in Series of Mathieu Functions

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Keywords:: Mathieu functions, expansions in series of
Notes:: See Arscott (1964b, §3.9.1).
Permalink:: http://dlmf.nist.gov/28.11
See also:: Annotations for Ch.28

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	$\displaystyle\alpha_{n}$	$\displaystyle=\frac{1}{\pi}\int_{0}^{2\pi}f(x)\operatorname{ce}_{n}\left(x,q% \right)\,\mathrm{d}x,$
28.11.2	$\displaystyle\beta_{n}$	$\displaystyle=\frac{1}{\pi}\int_{0}^{2\pi}f(x)\operatorname{se}_{n}\left(x,q% \right)\,\mathrm{d}x.$
ⓘ 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

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Keywords:: Mathieu functions, expansions in series of
See also:: Annotations for §28.11 and Ch.28

With the notation of §28.4,

28.11.3		$1=2\sum_{n=0}^{\infty}A_{0}^{2n}(q)\operatorname{ce}_{2n}\left(z,q\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.11.E3 Encodings: TeX, pMML, png See also: Annotations for §28.11, §28.11 and Ch.28

28.11.4		$\cos 2mz=\sum_{n=0}^{\infty}A_{2m}^{2n}(q)\operatorname{ce}_{2n}\left(z,q% \right),$
$m\neq 0$ ,
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.11.E4 Encodings: TeX, pMML, png See also: Annotations for §28.11, §28.11 and Ch.28

28.11.5	$\displaystyle\cos(2m+1)z$	$\displaystyle=\sum_{n=0}^{\infty}A_{2m+1}^{2n+1}(q)\operatorname{ce}_{2n+1}% \left(z,q\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.11.E5 Encodings: TeX, pMML, png See also: Annotations for §28.11, §28.11 and Ch.28
28.11.6	$\displaystyle\sin(2m+1)z$	$\displaystyle=\sum_{n=0}^{\infty}B_{2m+1}^{2n+1}(q)\operatorname{se}_{2n+1}% \left(z,q\right),$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.11.E6 Encodings: TeX, pMML, png See also: Annotations for §28.11, §28.11 and Ch.28
28.11.7	$\displaystyle\sin(2m+2)z$	$\displaystyle=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)\operatorname{se}_{2n+2}% \left(z,q\right).$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.11.E7 Encodings: TeX, pMML, png See also: Annotations for §28.11, §28.11 and Ch.28