§28.11 Expansions in Series of Mathieu Functions

Notes:: See Arscott (1964b, §3.9.1).
Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.11

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 $f(z)=\alpha _{0}\mathop{\mathrm{ce}_{{0}}\/}\nolimits\!\left(z,q\right)+\sum _{{n=1}}^{\infty}\left(\alpha _{n}\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)+\beta _{n}\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $f(z)$ : function, $\alpha _{n}$ and $\beta _{n}$
Referenced by:: §28.11
Permalink:: http://dlmf.nist.gov/28.11.E1
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where

28.11.2

$\alpha _{n}=\frac{1}{\pi}\int _{0}^{{2\pi}}f(x)\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(x,q\right)dx,$

$\beta _{n}=\frac{1}{\pi}\int _{0}^{{2\pi}}f(x)\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(x,q\right)dx.$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $dx$ : differential of $x$ , $\int$ : integral, $q=h^{2}$ : parameter, $n$ : integer, $x$ : real variable, $f(z)$ : function, $\alpha _{n}$ and $\beta _{n}$
Permalink:: http://dlmf.nist.gov/28.11.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

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

Keywords:: Mathieu functions

With the notation of §28.4,

28.11.3 $1=2\sum _{{n=0}}^{{\infty}}A_{0}^{{2n}}(q)\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(z,q\right),$

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

28.11.4 $\mathop{\cos\/}\nolimits 2mz=\sum _{{n=0}}^{{\infty}}A_{{2m}}^{{2n}}(q)\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(z,q\right),$ $m\neq 0$ ,

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

28.11.5 $\mathop{\cos\/}\nolimits(2m+1)z=\sum _{{n=0}}^{{\infty}}A_{{2m+1}}^{{2n+1}}(q)\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits\!\left(z,q\right),$

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

28.11.6 $\mathop{\sin\/}\nolimits(2m+1)z=\sum _{{n=0}}^{{\infty}}B_{{2m+1}}^{{2n+1}}(q)\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits\!\left(z,q\right),$

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

28.11.7 $\mathop{\sin\/}\nolimits(2m+2)z=\sum _{{n=0}}^{{\infty}}B_{{2m+2}}^{{2n+2}}(q)\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits\!\left(z,q\right).$

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