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28 Mathieu Functions and Hill’s Equation Mathieu Functions of Noninteger Order 28.18 Integrals and Integral Equations 28.20 Definitions and Basic Properties

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

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

Symbols:: : base of exponential function, : complex variable, $\nu$ : complex parameter and : 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, : integer, : complex variable, $\nu$ : complex parameter, : 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, : differential of , $\int$ : integral, $q=h^{2}$ : parameter, : integer, : complex variable, $\nu$ : complex parameter, : 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 .

¶ 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, : base of exponential function, $q=h^{2}$ : parameter, : integer, : 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.

© 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. 28.18 Integrals and Integral Equations 28.20 Definitions and Basic Properties