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## 6.9 Mathieu Functions

Notation: k2 = q. For definition of the coefficients Ap(m) and Bp(m) see section 8.6.

### 6.91 Mathieu functions

6.911

1.

$\underset{0}{\overset{2\pi }{\int }}\text{c}{e}_{m}\left(z,q\right)\text{c}{e}_{p}\left(z,q\right)\mathrm{d}z=0$

MA

$[m≠p]$

2.

$\underset{0}{\overset{2\pi }{\int }}c{e}_{2n}\left(z,q{\right)\right]}^{2}\mathrm{d}z=2\pi {\left[{A}_{0}^{\left(2n\right)}\right]}^{2}+\pi \sum _{r=1}^{\infty }{\left[{A}_{2r}^{\left(2n\right)\right]}\right]}^{2}=\pi$

MA

3.

$\begin{array}{l}\underset{0}{\overset{2\pi }{\int }}{\left[\text{c}{\text{e}}_{2n+1}\left(z,q\right)\right]}^{2}\text{d}z=\pi \sum _{r=0}^{\infty }{\left[{A}_{2r+1}^{\left(2n+1\right)}\right]}^{2}=\pi \end{array}$

ET II 411(50)

4.

$\begin{array}{l}\underset{0}{\overset{2\pi }{\int }}\left[{\text{se}}_{m}\left(z,q\right){\text{se}}_{p}\text{(}z,q\right)\text{d}z=0\end{array}$

ET II 411(50) ...

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