Formula:4117

Digital Library of Mathematical Functions ID 10.22.E46

${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at\right)}{(t^{2}+b^{2})^{\mu+1}}\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\Gamma\left(\mu+1\right)}K_{\nu-\mu}\left(ab\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\Re b>0}}$
${\displaystyle{\displaystyle-1<\Re\nu<2\Re\mu+\tfrac{3}{2}}}$
${\displaystyle{\displaystyle a>0}}$
${\displaystyle{\displaystyle\Re b>0}}$
${\displaystyle{\displaystyle-1<\Re\nu<2\Re\mu+\tfrac{3}{2}}}$

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
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the second kind
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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