# Mathieu functions

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##### 1: 28.12 Definitions and Basic Properties
###### §28.12(ii) Eigenfunctions $\operatorname{me}_{\nu}\left(z,q\right)$
They have the following pseudoperiodic and orthogonality properties: …
##### 3: 28.20 Definitions and Basic Properties
###### §28.20(vii) Shift of Variable
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …
##### 5: 28.33 Physical Applications
###### §28.33 Physical Applications
Mathieu functions occur in practical applications in two main categories: …
###### §28.33(iii) Stability and Initial-Value Problems
• Torres-Vega et al. (1998) for Mathieu functions in phase space.

• ##### 6: Gerhard Wolf
Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. His book Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies (with J. …
##### 8: 28.22 Connection Formulas
###### §28.22 Connection Formulas
The joining factors in the above formulas are given by …
$\operatorname{ge}_{m}\left(0,h^{2}\right)=\tfrac{1}{2}\pi S_{m}(h^{2})\left(g_% {\mathit{o},m}(h)\right)^{2}\operatorname{se}_{m}'\left(0,h^{2}\right).$
28.22.13 ${\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)=\frac{{\operatorname{M}^{(1)}_{% \nu}}\left(0,h\right)}{\operatorname{me}_{\nu}\left(0,h^{2}\right)}% \operatorname{Me}_{\nu}\left(z,h^{2}\right).$
##### 9: 28.11 Expansions in Series of Mathieu Functions
###### §28.11 Expansions in Series of MathieuFunctions
28.11.1 $f(z)=\alpha_{0}\operatorname{ce}_{0}\left(z,q\right)+\sum_{n=1}^{\infty}\left(% \alpha_{n}\operatorname{ce}_{n}\left(z,q\right)+\beta_{n}\operatorname{se}_{n}% \left(z,q\right)\right),$
28.11.3 $1=2\sum_{n=0}^{\infty}A_{0}^{2n}(q)\operatorname{ce}_{2n}\left(z,q\right),$
28.11.4 $\cos 2mz=\sum_{n=0}^{\infty}A_{2m}^{2n}(q)\operatorname{ce}_{2n}\left(z,q% \right),$ $m\neq 0$,
28.11.7 $\sin(2m+2)z=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)\operatorname{se}_{2n+2}\left% (z,q\right).$