§25.13 Periodic Zeta Function

Notes:: See Apostol (1976, pp. 257 and 273).
Keywords:: Hurwitz zeta function, periodic zeta function, polylogarithms
Permalink:: http://dlmf.nist.gov/25.13

The notation $\mathop{F\/}\nolimits\!\left(x,s\right)$ is used for the polylogarithm $\mathop{\mathrm{Li}_{{s}}\/}\nolimits\!\left(e^{{2\pi ix}}\right)$ with real:

25.13.1 $\mathop{F\/}\nolimits\!\left(x,s\right)=\sum_{{n=1}}^{\infty}\frac{e^{{2\pi inx% }}}{n^{s}},$

Defines:: $\mathop{F\/}\nolimits\!\left(x,s\right)$ : periodic zeta function
Symbols:: : base of exponential function, : nonnegative integer, : real variable and : complex variable
Permalink:: http://dlmf.nist.gov/25.13.E1
Encodings:: TeX, pMML, png

where $\realpart{s}>1$ if is an integer, $\realpart{s}>0$ otherwise.

$\mathop{F\/}\nolimits\!\left(x,s\right)$ is periodic in with period 1, and equals $\mathop{\zeta\/}\nolimits\!\left(s\right)$ when is an integer. Also,

25.13.2 $\mathop{F\/}\nolimits\!\left(x,s\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(1-s\right)}{(2\pi)^{{1-s}}}\*\left(e^{{\pi i(1-s)/2}}\mathop{\zeta\/}% \nolimits\!\left(1-s,x\right)+e^{{\pi i(s-1)/2}}\mathop{\zeta\/}\nolimits\!% \left(1-s,1-x\right)\right),$

, $\realpart{s}>1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, : base of exponential function, $\mathop{F\/}\nolimits\!\left(x,s\right)$ : periodic zeta function, $\realpart{}$ : real part, : real variable and : complex variable
Permalink:: http://dlmf.nist.gov/25.13.E2
Encodings:: TeX, pMML, png

25.13.3 $\mathop{\zeta\/}\nolimits\!\left(1-s,x\right)=\frac{\mathop{\Gamma\/}\nolimits% \!\left(s\right)}{(2\pi)^{s}}\left(e^{{-\pi is/2}}\mathop{F\/}\nolimits\!\left% (x,s\right)+e^{{\pi is/2}}\mathop{F\/}\nolimits\!\left(-x,s\right)\right),$

, $\realpart{s}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, : base of exponential function, $\mathop{F\/}\nolimits\!\left(x,s\right)$ : periodic zeta function, $\realpart{}$ : real part, : real variable and : complex variable
Permalink:: http://dlmf.nist.gov/25.13.E3
Encodings:: TeX, pMML, png

© 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. 25.12 Polylogarithms 25.14 Lerch’s Transcendent