28 Mathieu Functions and Hill’s Equation Mathieu Functions of Noninteger Order 28.13 Graphics 28.15 Expansions for Small

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§28.14 Fourier Series

ⓘ

Keywords:: Fourier coefficients, Fourier series, Mathieu functions, normalization, recurrence relations
Notes:: See Meixner and Schäfke (1954, pp. 111, 115, and 121).
Referenced by:: §28.19, §28.23, §28.28(ii)
Permalink:: http://dlmf.nist.gov/28.14
See also:: Annotations for Ch.28

The Fourier series

28.14.1	$\displaystyle\operatorname{me}_{\nu}\left(z,q\right)$	$\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)e^{\mathrm{i}(\nu+2m)z},$
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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	$\displaystyle\operatorname{ce}_{\nu}\left(z,q\right)$	$\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)\cos(\nu+2m)z,$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, $m$ : integer, $q=h^{2}$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients Permalink: http://dlmf.nist.gov/28.14.E2 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28
28.14.3	$\displaystyle\operatorname{se}_{\nu}\left(z,q\right)$	$\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)\sin(\nu+2m)z,$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, $m$ : integer, $q=h^{2}$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients Permalink: http://dlmf.nist.gov/28.14.E3 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28

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

28.14.4		${qc_{2m+2}-\left(a-(\nu+2m)^{2}\right)c_{2m}+qc_{2m-2}=0},$
$a=\lambda_{\nu}\left(q\right),c_{2m}=c_{2m}^{\nu}(q)$ ,
ⓘ Symbols: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter, $a$ : parameter and $c_{2m}(q)$ : coefficients Referenced by: item (d), item (d) Permalink: http://dlmf.nist.gov/28.14.E4 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28

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		$\frac{c^{\nu}_{2m}(q)}{c^{\nu}_{2m\mp 2}(q)}=\frac{-q}{4m^{2}}\left(1+O\left(% \frac{1}{m}\right)\right),$
$m\to\pm\infty$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients Permalink: http://dlmf.nist.gov/28.14.E6 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28

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		$c_{2m}^{\nu}(q)=\left(\frac{(-1)^{m}q^{m}\Gamma\left(\nu+1\right)}{m!\,2^{2m}% \Gamma\left(\nu+m+1\right)}+O\left(q^{m+2}\right)\right)c_{0}^{\nu}(q).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients A&S Ref: 20.3.16 Permalink: http://dlmf.nist.gov/28.14.E10 Encodings: TeX, pMML, png See also: Annotations for §28.14 and Ch.28

28.13 Graphics 28.15 Expansions for Small

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