§25.14 Lerch’s Transcendent

Permalink:: http://dlmf.nist.gov/25.14

§25.14(i) Definition
§25.14(ii) Properties

§25.14(i) Definition

Keywords:: Hurwitz zeta function, Lerch’s transcendent, polylogarithms
Permalink:: http://dlmf.nist.gov/25.14.i

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

Defines:: $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ : Lerch’s transcendent
Symbols:: $\realpart{}$ : real part, : nonnegative integer, : real or complex parameter, : complex variable and : complex variable
Referenced by:: §25.14(ii)
Permalink:: http://dlmf.nist.gov/25.14.E1
Encodings:: TeX, pMML, png

For other values of , $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ 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 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\mathop{\Phi\/}\nolimits\!\left(1,% s,a\right),$ $\realpart{s}>1$ , $a\neq 0,-1,-2,\dots$ ,

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ : Lerch’s transcendent, $\realpart{}$ : real part, : real or complex parameter and : complex variable
Permalink:: http://dlmf.nist.gov/25.14.E2
Encodings:: TeX, pMML, png

25.14.3 $\mathop{\mathrm{Li}_{{s}}\/}\nolimits\!\left(z\right)=z\mathop{\Phi\/}% \nolimits\!\left(z,s,1\right),$ $\realpart{s}>1$ , $|z|\leq 1$ .

Symbols:: $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ : Lerch’s transcendent, $\mathop{\mathrm{Li}_{{s}}\/}\nolimits\!\left(z\right)$ : polylogarithm, $\realpart{}$ : real part, : complex variable and : complex variable
Permalink:: http://dlmf.nist.gov/25.14.E3
Encodings:: TeX, pMML, png

§25.14(ii) Properties

Keywords:: Lerch’s transcendent
Permalink:: http://dlmf.nist.gov/25.14.ii

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

25.14.4 $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)=z^{m}\mathop{\Phi\/}\nolimits\!% \left(z,s,a+m\right)+\sum_{{n=0}}^{{m-1}}\frac{z^{n}}{(a+n)^{s}}.$

Symbols:: $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ : Lerch’s transcendent, : nonnegative integer, : nonnegative integer, : real or complex parameter, : complex variable and : complex variable
Permalink:: http://dlmf.nist.gov/25.14.E4
Encodings:: TeX, pMML, png

25.14.5 $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)=\frac{1}{\mathop{\Gamma\/}% \nolimits\!\left(s\right)}\int_{0}^{\infty}\frac{x^{{s-1}}e^{{-ax}}}{1-ze^{{-x% }}}dx,$ $\realpart{s}>0$ , $\realpart{a}>0$ , $z\in\Complex\setminus[1,\infty)$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ : Lerch’s transcendent, : half-closed interval, $\Complex$ : complex plane (excluding infinity), : differential of , $\in$ : element of, : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $\setminus$ : set subtraction, : real variable, : real or complex parameter, : complex variable and : complex variable
Permalink:: http://dlmf.nist.gov/25.14.E5
Encodings:: TeX, pMML, png

25.14.6 $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)=\frac{1}{2}a^{{-s}}+\int_{0}^{% \infty}\frac{z^{x}}{(a+x)^{s}}dx-2\int_{0}^{\infty}\frac{\mathop{\sin\/}% \nolimits\!\left(x\mathop{\ln\/}\nolimits z-s\mathop{\mathrm{arctan}\/}% \nolimits\!\left(x/a\right)\right)}{(a^{2}+x^{2})^{{s/2}}(e^{{2\pi x}}-1)}dx,$ $\realpart{s}>0$ if

; $\realpart{s}>1$ if $|z|=1,\realpart{a}>0$ .

Symbols:: $\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$ : Lerch’s transcendent, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : real or complex parameter, : complex variable and : complex variable
Permalink:: http://dlmf.nist.gov/25.14.E6
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 25.13 Periodic Zeta Function 25.15 Dirichlet $\mathop{L\/}\nolimits$ -functions

§25.14 Lerch’s Transcendent

Contents

§25.14(i) Definition

§25.14(ii) Properties