Formula:5170

Digital Library of Mathematical Functions ID 13.10.E14

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

Constraint(s)

${\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}$
${\displaystyle{\displaystyle x>0}}$
${\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}$

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
• ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}$: 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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