Formula:3224

Digital Library of Mathematical Functions ID 8.6.E12

${\displaystyle{\displaystyle\Gamma\left(a,z\right)=-\frac{z^{a-1}e^{-z}}{\Gamma\left(1-a\right)}\*\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+1-a\right)\frac{\pi z^{-s}}{\sin\left(\pi s\right)}\mathrm{d}s,}}$

Constraint(s)

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

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma 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
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}$: incomplete gamma function
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: parameter
• ${\displaystyle{\displaystyle c}}$: real constant
 

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