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

§28.14 Fourier Series

Notes:: See Meixner and Schäfke (1954, pp. 111, 115, and 121).
Keywords:: Fourier series, Mathieu functions
Referenced by:: ¶ ‣ §28.19, §28.23, §28.28(ii)
Permalink:: http://dlmf.nist.gov/28.14

The Fourier series

28.14.1 $\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(z,q\right)=\sum_{{m=-\infty}}^{% {\infty}}c^{{\nu}}_{{2m}}(q)e^{{i(\nu+2m)z}},$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : base of exponential function, : 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

28.14.2 $\mathop{\mathrm{ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)=\sum_{{m=-\infty}}^{% {\infty}}c^{{\nu}}_{{2m}}(q)\mathop{\cos\/}\nolimits(\nu+2m)z,$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : 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

28.14.3 $\mathop{\mathrm{se}_{{\nu}}\/}\nolimits\!\left(z,q\right)=\sum_{{m=-\infty}}^{% {\infty}}c^{{\nu}}_{{2m}}(q)\mathop{\sin\/}\nolimits(\nu+2m)z,$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sin\/}\nolimits z$ : 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

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

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

Symbols:: $\mathop{\lambda_{{\nu+2n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter, : parameter and $c_{{2m}}(q)$ : coefficients
Referenced by:: §28.34(ii), §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.14.E4
Encodings:: TeX, pMML, png

and the normalization relation

28.14.5 $\sum_{{m=-\infty}}^{{\infty}}\left(c_{{2m}}^{{\nu}}(q)\right)^{2}=1;$

Symbols:: : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{{2m}}(q)$ : coefficients
Referenced by:: §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.14.E5
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : 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

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

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

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

28.14.8 $c_{{2m}}^{{\nu}}(-q)=(-1)^{m}c_{{2m}}^{{\nu}}(q).$

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

When ,

28.14.9

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

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

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

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : : factorial, : 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

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