DLMF:10.22.E48

MaRDI QIDQ4119

• Label: DLMF:10.22.E48

Digital Library of Mathematical Functions ID 10.22.E48

${\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(x\cosh\phi\right)(\cosh\phi)^{1-\mu}(\sinh\phi)^{2\nu+1}\mathrm{d}\phi=2^{\nu}\Gamma\left(\nu+1\right)x^{-\nu-1}J_{\mu-\nu-1}\left(x\right),}}$

Constraint(s)

${\displaystyle{\displaystyle x>0,\Re\nu>-1,\Re\mu>2\Re\nu+\tfrac{1}{2}}}$

Symbols List

• ${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the first kind
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\cosh\NVar{z}}}$: hyperbolic cosine function
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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