Formula:5074

Digital Library of Mathematical Functions ID 13.4.E14

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

Constraint(s)

${\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi}}$
${\displaystyle{\displaystyle a\neq 1,2,3,\dots}}$
${\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\frac{1}{2}\pi}}$

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
• ${\displaystyle{\displaystyle\pi}}$: 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
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle z}}$: complex variable
 

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