Formula:DLMF:25.14:E5

<< Formula:DLMF:25.14:E4

formula in Lerch's Transcendent

Formula:DLMF:25.14:E6 >>

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

Constraint(s)

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

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Phi}}}$ : Lerch's transcendent : http://dlmf.nist.gov/25.14#E1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle\in}}}$ : element of : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9
${\displaystyle{\displaystyle{\displaystyle\mathbb{C}}}}$ : set of complex numbers : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1
${\displaystyle{\displaystyle{\displaystyle\setminus}}}$ : set subtraction : http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19

Bibliography

Equation (5), Section 25.14 of DLMF.

URL links

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<< Formula:DLMF:25.14:E4

formula in Lerch's Transcendent

Formula:DLMF:25.14:E6 >>