Formula:9166

Digital Library of Mathematical Functions ID 28.28.E43

${\displaystyle{\displaystyle\widehat{\beta}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\sin t\mathrm{se}_{n}\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{p}\dfrac{2}{\mathrm{i}\pi}\dfrac{\mathrm{se}_{n}'\left(0,h^{2}\right)\mathrm{ce}_{m}\left(0,h^{2}\right)}{h\mathrm{Dsc}_{1}\left(n,m,0\right)}.}}$

Symbols List

• ${\displaystyle{\displaystyle\mathrm{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)}}$: cross-products of radial Mathieu functions and their derivatives
• ${\displaystyle{\displaystyle\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)}}$:
• ${\displaystyle{\displaystyle\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle m}}$: integer
• ${\displaystyle{\displaystyle h}}$: parameter
• ${\displaystyle{\displaystyle n}}$: integer
 

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