From: israel@math.ubc.ca (Robert Israel)
Subject: Re: integrate (x^b/(1-x^a))
Date: 18 Jan 2000 21:26:09 GMT
Newsgroups: sci.math
Summary: [missing]

In article <388425D3.763BF041@teleweb.at>,
 Helmut Kahovec <helmut.kahovec@teleweb.at> writes:

> The Lerch Phi function is defined as follows: 

>                     infinity
>                      ------     n
>                       \        z
>  LerchPhi(z,a,v) =     >    -------      
>                       /           a
>                      ------  (v+n)
>                      n = 0 
 
> This definition is valid for abs(z)<1. By analytic continuation, it is
> extended to the whole complex plane. Singularities are encountered for:

>  (1) z=1 and a=0 or a=1
>  (2) v a non-positive integer and Re(a) also non-positive

This Maple help page has several errors.  In general, z=1 is a branch point
(except if a is a non-positive integer).  The branch cut is taken for 
z in [1,infinity).

z=1 is always a singularity: I think what (1) is trying to say is that 
the series converges at z=1 if Re(a) > 1 (and v is not a non-positive integer).
 
(2) should say: v a non-positive integer and Re(a) non-negative.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2