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Topic: ISO Algorithm for Lerch transcendent (polylog).
Replies: 0

 Martin Johansson Posts: 4 Registered: 12/7/04
ISO Algorithm for Lerch transcendent (polylog).
Posted: Dec 9, 1996 8:44 PM

I'm looking for an algorithm to calculate the "Lerch transcendent"
(as defined in [1], p. 744) which is given by

\Phi(z,s,a) = \sum_{k=0}^{\infty}z^k/(a+k)^s. (1)

The brute-force implementation of the sum works but is very slowly
convergent for s in certain intervals. Olver ([2], pp. 296--299)
considers the similar function \Phi(exp(i*beta),s,0) with the summation
starting at k=1 (denoted "polylogarithm function" in [1]). He expresses
the sum in terms of an integral in an exercise.

Q1: Does anyone know of a readily available algorithm for
evaluating the Lerch transcendent "fast"? A search through the
mathematical resources on the net comes up with a few algorithms
but only for real-valued z (I have z=exp(i*...)).

Q2: I have considered doing a series expansion of (1) for k->Infinity
and then using the integral representation of Olver; could this
be advantageous in terms of speed?

Q3: Better yet, has anybody implemented this in Matlab? :-)

Any help or pointers appreciated! TIA,

mj

Refs:
[1] S. Wolfram, "The Mathematica Book, 3rd ed."
[2] F. W. J. Olver, "Asymptotics and Special Functions"