Ξ¦ β‘ ( z , s , a ) = β n = 0 β z n ( a + n ) s Lerch-Phi π§ π π superscript subscript π 0 superscript π§ π superscript π π π {\displaystyle{\displaystyle{\displaystyle{\Phi\left(z,s,a\right)=\sum_{n=0}^{% \infty}\frac{z^{n}}{(a+n)^{s}}}}}} {\displaystyle { \LerchPhi@{z}{s}{a} = \sum_{n=0}^\infty \frac{z^n}{(a+n)^s} } }
\HurwitzZeta β’ @ β’ s β’ a = Ξ¦ β‘ ( 1 , s , a ) \HurwitzZeta @ π π Lerch-Phi 1 π π {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\Phi\left(1,s,a% \right)}}} {\displaystyle \HurwitzZeta@{s}{a} = \LerchPhi@{1}{s}{a} }
\Polylogarithm β’ s β’ @ β’ z = z β’ Ξ¦ β‘ ( z , s , 1 ) \Polylogarithm π @ π§ π§ Lerch-Phi π§ π 1 {\displaystyle{\displaystyle{\displaystyle\Polylogarithm{s}@{z}=z\Phi\left(z,s% ,1\right)}}} {\displaystyle \Polylogarithm{s}@{z} = z \LerchPhi@{z}{s}{1} }
Ξ¦ β‘ ( z , s , a ) = z m β’ Ξ¦ β‘ ( z , s , a + m ) + β n = 0 m - 1 z n ( a + n ) s Lerch-Phi π§ π π superscript π§ π Lerch-Phi π§ π π π superscript subscript π 0 π 1 superscript π§ π superscript π π π {\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}}}}} {\displaystyle \LerchPhi@{z}{s}{a} = z^m \LerchPhi@{z}{s}{a+m} + \sum_{n=0}^{m-1} \frac{z^n}{(a+n)^s} }
Ξ¦ β‘ ( z , s , a ) = 1 Ξ β‘ ( s ) β’ β« 0 β x s - 1 β’ e - a β’ x 1 - z β’ e - x β’ d x Lerch-Phi π§ π π 1 Euler-Gamma π superscript subscript 0 superscript π₯ π 1 π π₯ 1 π§ π₯ π₯ {\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}}} {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-z\expe^{-x}} \diff{x} }
Ξ¦ β‘ ( z , s , a ) = 1 2 β’ a - s + β« 0 β z x ( a + x ) s β’ d x - 2 β’ β« 0 β sin β‘ ( x β’ ln β‘ z - s β’ arctan β‘ ( x / a ) ) ( a 2 + x 2 ) s / 2 β’ ( e 2 β’ Ο β’ x - 1 ) β’ d x Lerch-Phi π§ π π 1 2 superscript π π superscript subscript 0 superscript π§ π₯ superscript π π₯ π π₯ 2 superscript subscript 0 π₯ π§ π π₯ π superscript superscript π 2 superscript π₯ 2 π 2 2 π₯ 1 π₯ {\displaystyle{\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{2}a^% {-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\mathrm{d}x-2\int_{0}^{\infty}% \frac{\sin\left(x\ln z-s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x% ^{2})^{s/2}({\mathrm{e}^{2\pi x}}-1)}\mathrm{d}x}}} {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{2} a^{-s} + \int_0^\infty \frac{z^x}{(a+x)^s} \diff{x} - 2 \int_0^\infty \frac{\sin@{x \ln@@{z} - s \atan@{x/a}}} {(a^2 + x^2)^{s/2} (\expe^{2 \cpi x} - 1)} \diff{x} }