§28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions

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Keywords:: Mathieu functions, expansions in series of
Notes:: See Meixner and Schäfke (1954, §2.28).
Referenced by:: §28.30(ii)
Permalink:: http://dlmf.nist.gov/28.19
See also:: Annotations for Ch.28

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		$f(z+\pi)=e^{\mathrm{i}\nu\pi}f(z).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $\nu$ : complex parameter and $f(z)$ : function Permalink: http://dlmf.nist.gov/28.19.E1 Encodings: TeX, pMML, png See also: Annotations for §28.19 and Ch.28

Then

28.19.2		$f(z)=\sum_{n=-\infty}^{\infty}f_{n}\operatorname{me}_{\nu+2n}\left(z,q\right),$
ⓘ Defines: $f(z)$ : function (locally) Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 and Ch.28

where

28.19.3		$f_{n}=\frac{1}{\pi}\int_{0}^{\pi}f(z)\operatorname{me}_{\nu+2n}\left(-z,q% \right)\,\mathrm{d}z.$
ⓘ Defines: $f_{n}$ : coefficients (locally) Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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, $z$ : complex variable, $\nu$ : complex parameter and $f(z)$ : function Permalink: http://dlmf.nist.gov/28.19.E3 Encodings: TeX, pMML, png See also: Annotations for §28.19 and Ch.28

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

Example

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See also:: Annotations for §28.19 and Ch.28

28.19.4		$e^{\mathrm{i}\nu z}=\sum_{n=-\infty}^{\infty}c^{\nu+2n}_{-2n}(q)\operatorname{% me}_{\nu+2n}\left(z,q\right),$
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients Permalink: http://dlmf.nist.gov/28.19.E4 Encodings: TeX, pMML, png See also: Annotations for §28.19, §28.19 and Ch.28

where the coefficients are as in §28.14.

§28.19 Expansions in Series of meν+2⁢n Functions

Example

§28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions