Formula:5162

Digital Library of Mathematical Functions ID 13.10.E6

${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}\left(a,b,t^{2}\right)\mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-\tfrac{1}{2}\right)U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\Re b>\tfrac{1}{2}}}$
${\displaystyle{\displaystyle\Re z>0}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: 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
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle z}}$: complex variable
 

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