Formula:9148

Digital Library of Mathematical Functions ID 28.28.E29

${\displaystyle{\displaystyle\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\mathrm{me}_{\nu}'\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}\mathrm{D}_{1}\left(\nu,\nu+2m+1,z\right),}}$

Symbols List

• ${\displaystyle{\displaystyle\mathrm{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)}}$: cross-products of modified Mathieu functions and their derivatives
• ${\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\cosh\NVar{z}}}$: hyperbolic cosine function
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle m}}$: integer
• ${\displaystyle{\displaystyle h}}$: parameter
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
• ${\displaystyle{\displaystyle\alpha^{(s)}_{\nu,m}}}$: 28.28.E27
 

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