28 Mathieu Functions and Hill’s Equation Mathieu Functions of Integer Order 28.10 Integral Equations 28.12 Definitions and Basic Properties

§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 be a $2\pi$ -periodic function that is analytic in an open doubly-infinite strip that contains the real axis, and 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, : integer, : complex variable, : function, $\alpha_{n}$ and $\beta_{n}$
Referenced by:: §28.11
Permalink:: http://dlmf.nist.gov/28.11.E1
Encodings:: TeX, pMML, png

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, : differential of , $\int$ : integral, $q=h^{2}$ : parameter, : integer, : real variable, : 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 . See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of 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, : integer, : 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, : integer, $q=h^{2}$ : parameter, : integer, : 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, : integer, $q=h^{2}$ : parameter, : integer, : 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, : integer, $q=h^{2}$ : parameter, : integer, : 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, : integer, $q=h^{2}$ : parameter, : integer, : complex variable and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.11.E7
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.10 Integral Equations 28.12 Definitions and Basic Properties