Re: Polylogarithm Evaluation

alonsoyjavier@xxxxxxxxx wrote:
On 12 nov, 19:03, Paul J Gans <g...@xxxxxxxxx> wrote:
Defining a polylogarithm by:

g(x,z) = \sum_{n=1}^\infty z^n/n^x

where all quantities are real, can anyone point me
to a reasonable method for the computation of g when
z is less than but very close to 1? In the problems
I am interested in z = 3/2 or 1/2 and 0 \le x \le 1

This is, of course, \zeta(x) when z = 1.

These functions occur in the discussion of the condensation
of an ideal Bose-Einstein gas.

Thanks in advance.

--
--- Paul J. Gans

Paul,
this function is a special case of Lerch's transcendent, which has an
expansion in powers of log(z).
See Erd?lyi et al., Bateman Project Vol. I, p. 29. If you don't have
this reference, see
http://en.wikipedia.org/wiki/Lerch_transcendent

You mention the value z = 3/2; at this point the function can be
defined through analytic continuation.

Thank you very much. The wiki page provided a link both
to what looks like a decent computational method with
written explanation and to a compilation of programes,
including the Lerch function, written in C, which is
exactly what I wanted.

--
--- Paul J. Gans
.