Formula:5066

Digital Library of Mathematical Functions ID 13.4.E6

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

Constraint(s)

${\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}$
${\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}}$
${\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}$
${\displaystyle{\displaystyle n=0,1,2,\dots}}$
${\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}}$

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
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle z}}$: complex variable
 

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