Table of Integrals, Series, and Products, 8th Edition

6.9 Mathieu Functions

Notation: k² = q. For definition of the coefficients A_p^(m) and B_p^(m) see section 8.6.

6.91 Mathieu functions

6.911

1. 

$\underset{0}{\overset{2\pi }{{\displaystyle \int }}}\text{c}{e}_m(z,q)\text{c}{e}_p(z,q)\mathrm{d}z=0$

  MA

$[m\ne p]$

2. 

$\underset{0}{\overset{2\pi }{{\displaystyle \int }}}c{e}_{2n}(z,q{)]}^2\mathrm{d}z=2\pi {[A_0^{(2n)}]}^2+\pi {\displaystyle \sum }_{r=1}^{\infty }{\left[A_{2r}^{(2n)]}\right]}^2=\pi$

  MA

3. 

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

  ET II 411(50)

4. 

$\begin{array}{l}{\displaystyle \underset{0}{\overset{2\pi }{\int }}[{\text{se}}_m(z,q){\text{se}}_p\text{(}z,q)\text{d}z=0}\end{array}$

  ET II 411(50) ...