Lerch's Transcendent

<< Periodic Zeta Function

Lerch's Transcendent

Dirichlet L-functions >>

Definition

${\displaystyle{\displaystyle{\displaystyle{\Phi\left(z,s,a\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0,-1,-2,\dots,|z|<1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{s}>1,|z|=1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\Phi\left(1,s,a\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a\neq 0,-1,-2,\dots}}}$

${\displaystyle{\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=z\Phi\left(z,s,1\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ &
${\displaystyle{\displaystyle{\displaystyle|z|\leq 1}}}$

Properties

${\displaystyle{\displaystyle{\displaystyle\Phi\left(z,s,a\right)=z^{m}\Phi\left(z,s,a+m\right)+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0,-1,-2,\dots,|z|<1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{s}>1,|z|=1}}}$ &
${\displaystyle{\displaystyle{\displaystyle m=1,2,3,\dots}}}$

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

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

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

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

<< Periodic Zeta Function

Lerch's Transcendent

Dirichlet L-functions >>