# §25.14 Lerch’s Transcendent

## §25.14(i) Definition

 25.14.1 ${\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{% (a+n)^{s}}},$ $a\neq 0,-1,-2,\dots,|z|<1$; $\Re{s}>1,|z|=1$. Defines: $\mathop{\Phi\/}\nolimits\!\left(\NVar{z},\NVar{s},\NVar{a}\right)$: 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)

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

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

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

## §25.14(ii) Properties

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

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