§28.19 Expansions in Series of $\mathop{\mathrm{me}_{{\nu+2n}}\/}\nolimits$ Functions

Notes:: See Meixner and Schäfke (1954, §2.28).
Keywords:: Mathieu functions
Referenced by:: §28.30(ii)
Permalink:: http://dlmf.nist.gov/28.19

Let $q$ be a normal value (§28.12(i)) with respect to $\nu$ , and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. Assume also

28.19.1 $f(z+\pi)=e^{{i\nu\pi}}f(z).$

Symbols:: $e$ : base of exponential function, $z$ : complex variable, $\nu$ : complex parameter and $f(z)$ : function
Permalink:: http://dlmf.nist.gov/28.19.E1
Encodings:: TeX, pMML, png

Then

28.19.2 $f(z)=\sum _{{n=-\infty}}^{{\infty}}f_{n}\mathop{\mathrm{me}_{{\nu+2n}}\/}\nolimits\!\left(z,q\right),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $f(z)$ : function and $f_{n}$ : coefficients
Referenced by:: §28.19
Permalink:: http://dlmf.nist.gov/28.19.E2
Encodings:: TeX, pMML, png

where

28.19.3 $f_{n}=\frac{1}{\pi}\int _{{0}}^{{\pi}}f(z)\mathop{\mathrm{me}_{{\nu+2n}}\/}\nolimits\!\left(-z,q\right)dz.$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $dx$ : differential of $x$ , $\int$ : integral, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $f(z)$ : function and $f_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/28.19.E3
Encodings:: TeX, pMML, png

The series (28.19.2) converges absolutely and uniformly on compact subsets within $S$ .

¶ Example

28.19.4 $e^{{i\nu z}}=\sum _{{n=-\infty}}^{{\infty}}c^{{\nu+2n}}_{{-2n}}(q)\mathop{\mathrm{me}_{{\nu+2n}}\/}\nolimits\!\left(z,q\right),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $e$ : base of exponential function, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $\nu$ : complex parameter and $c_{{2m}}(q)$ : coefficients
Permalink:: http://dlmf.nist.gov/28.19.E4
Encodings:: TeX, pMML, png

where the coefficients are as in §28.14.

§28.19 Expansions in Series of Functions

¶ Example

§28.19 Expansions in Series of $\mathop{\mathrm{me}_{{\nu+2n}}\/}\nolimits$ Functions