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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 bruteforce implementation of the sum works but is very slowly convergent for s in certain intervals. Olver ([2], pp. 296299) 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 realvalued 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"



