Formula:8437

Digital Library of Mathematical Functions ID 25.14.E5

${\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\mathrm{d}x,}}$

Constraint(s)

${\displaystyle{\displaystyle\Re s>0}}$
${\displaystyle{\displaystyle\Re a>0}}$
${\displaystyle{\displaystyle z\in\mathbb{C}\setminus[1,\infty)}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)}}$: Lerch’s transcendent
• ${\displaystyle{\displaystyle[\NVar{a},\NVar{b})}}$:
• ${\displaystyle{\displaystyle\mathbb{C}}}$:
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\in}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\setminus}}$:
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
• ${\displaystyle{\displaystyle s}}$: complex variable
• ${\displaystyle{\displaystyle z}}$: complex variable
 

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