# §25.14 Lerch’s Transcendent

## §25.14(i) Definition

 25.14.1 ${\Phi\left(z,s,a\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $a\neq 0,-1,-2,\dots,|z|<1$; $\Re s>1,|z|=1$. ⓘ Defines: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$: Lerch’s transcendent Symbols: $\Re$: real part, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Referenced by: §25.14(ii) Permalink: http://dlmf.nist.gov/25.14.E1 Encodings: TeX, pMML, png See also: Annotations for 25.14(i), 25.14 and 25

For other values of $z$, $\Phi\left(z,s,a\right)$ is defined by analytic continuation. This is the notation used in Erdélyi et al. (1953a, p. 27). Lerch (1887) used $\mathfrak{K}(a,x,s)=\Phi\left(e^{2\pi ix},s,a\right)$.

The Hurwitz zeta function $\zeta\left(s,a\right)$25.11) and the polylogarithm $\mathrm{Li}_{s}\left(z\right)$25.12(ii)) are special cases:

 25.14.2 $\zeta\left(s,a\right)=\Phi\left(1,s,a\right),$ $\Re s>1$, $a\neq 0,-1,-2,\dots$,
 25.14.3 $\mathrm{Li}_{s}\left(z\right)=z\Phi\left(z,s,1\right),$ $\Re s>1$, $|z|\leq 1$.

## §25.14(ii) Properties

With the conditions of (25.14.1) and $m=1,2,3,\dots$,

 25.14.4 $\Phi\left(z,s,a\right)=z^{m}\Phi\left(z,s,a+m\right)+\sum_{n=0}^{m-1}\frac{z^{% n}}{(a+n)^{s}}.$
 25.14.5 $\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^% {s-1}e^{-ax}}{1-ze^{-x}}\mathrm{d}x,$ $\Re s>0$, $\Re a>0$, $z\in\mathbb{C}\setminus[1,\infty)$.
 25.14.6 $\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{% s}}\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x\ln z-s\operatorname{arctan}% \left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x,$ $\Re s>0$ if $|z|<1$; $\Re s>1$ if $|z|=1,\Re a>0$.

For these and further properties see Erdélyi et al. (1953a, pp. 27–31).