Formula:3409

Digital Library of Mathematical Functions ID 8.19.E25

${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}t^{b-1}E_{p}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(b\right)(1+a)^{-b}}{p+b-1}\*F\left(1,b;p+b;a/(1+a)\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\Re a>-1}}$
${\displaystyle{\displaystyle\Re(p+b)>1}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle E_{\NVar{p}}\left(\NVar{z}\right)}}$: generalized exponential integral
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle a}}$: parameter
• ${\displaystyle{\displaystyle p}}$: parameter
 

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