Formula:5172

Digital Library of Mathematical Functions ID 13.10.E16

${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}U\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\Gamma\left(\nu-b+2\right)x^{\frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(a,a-b+\nu+2,x\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu}}$
${\displaystyle{\displaystyle x>0}}$
${\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu}}$

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
• ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}$: Kummer confluent hypergeometric function
• ${\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)}}$: Olver’s confluent hypergeometric function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle x}}$: real variable
 

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