Formula:4120

Digital Library of Mathematical Functions ID 10.22.E49

${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt\right)\mathrm{d}t=\frac{(\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+\nu\right)\*\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\Re(\mu+\nu)>0,\Re(a\pm ib)>0}}$

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
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: Olver’s hypergeometric function
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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