Formula:DLMF:25.14:E6

<< Formula:DLMF:25.14:E5

formula in Lerch's Transcendent

Formula:DLMF:25.15:E1 >>

${\displaystyle{\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\diffd x-2\int_{0}^{\infty}\frac{\sin\!\left(x\ln z-s\mathrm{arctan}\!\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(\expe^{2\cpi x}-1)}\diffd x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ if ${\displaystyle{\displaystyle{\displaystyle|z|<1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ if ${\displaystyle{\displaystyle{\displaystyle|z|=1,\realpart{a}>0}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Phi}}}$ : Lerch's transcendent : http://dlmf.nist.gov/25.14#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{arctan}}}}$ : inverse tangent function : http://dlmf.nist.gov/4.23#SS2.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (6), Section 25.14 of DLMF.

URL links

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<< Formula:DLMF:25.14:E5

formula in Lerch's Transcendent

Formula:DLMF:25.15:E1 >>