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

§28.14 Fourier Series

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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 $\operatorname{me}_{\nu}\left(z,q\right)=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(% q)e^{\mathrm{i}(\nu+2m)z},$

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Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : integer, $q=h^{2}$ : parameter, : 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 $\operatorname{ce}_{\nu}\left(z,q\right)=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(% q)\cos(\nu+2m)z,$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, : integer, $q=h^{2}$ : parameter, : 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 $\operatorname{se}_{\nu}\left(z,q\right)=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(% q)\sin(\nu+2m)z,$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, : integer, $q=h^{2}$ : parameter, : 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 -plane. The coefficients satisfy

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

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Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter, : 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 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$

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Symbols:: : 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 , and by continuity for other values of .

The rate of convergence is indicated by

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : 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 $\nu$ , , and ,

28.14.7 $c_{-2m}^{-\nu}(q)=c_{2m}^{\nu}(q),$

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Symbols:: : 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 $c_{2m}^{\nu}(-q)=(-1)^{m}c_{2m}^{\nu}(q).$

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Symbols:: : 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 ,

28.14.9

$c_{0}^{\nu}(0)=1,$

$c_{2m}^{\nu}(0)=0,$ $m\neq 0$ .

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Symbols:: : 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 ( $\geq 1$ ) and $\nu$ fixed,

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, : factorial (as in ), : 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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