Re: Re: Mathematica 3.0.0 bug in LerchPhi function

*To*: mathgroup at smc.vnet.net*Subject*: [mg9240] Re: [mg9190] Re: [mg9065] Mathematica 3.0.0 bug in LerchPhi function*From*: David Withoff <withoff>*Date*: Fri, 24 Oct 1997 01:00:54 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Kai Koehler wrote: > Thanks once more to Dave Withoff for the fast clarification of the > problem. Still there is a point which should be dealt with: > > There is no such thing like a second "common" definition of the Lerch > Phi function. > > Lerch in his article in Acta Mathematica XI (p. 19-24) gives the first > definition (already in the title of his paper). This is also the > definition which one finds in the usual textbooks and formula > collections like Whittaker-Watson or Gradshteyn-Ryzhik. > > The second definition does not verify the functional relations of the > Lerch Phi function. Furthermore, it has a branching point of order two > in the variable a in contrast to the correct definition, which is > meromorphic in a and has no branching point at all. It is a very > strange idea to complicate matters by artificially creating a > branching point in a meromorphic function in a program like Mathematica > which has traditionally lots of problems with branching points and > Riemann surfaces. > > I have been told by Dave Withoff that the Hurwitz zeta function is > evaluated in Mathematica 3.0 in the same wrong way, namely as > > Sum[1/((k + a)^2)^(s/2), {k, 0, Infinity}]. > > I want to emphasize once more that Wolfram should provide free bug fixes > for serious problems like these. Also, Wolfram should officially inform > its customers about these errors, e.g. on their web page. > > Best regards > > Kai Koehler Since the conclusion will depend on the definition of "common", it probably isn't useful to get into a debate about whether or not the Mathematica definitions of LerchPhi and Zeta are "common" or not. Probably the most useful way to pursue this would be for some group of active users of these functions to review the different definitions of LerchPhi and Zeta, document their conclusions, and incorporate those conclusions into Mathematica. If that discussion leads to some changes in Mathematica, that would be fine with me. On the other hand, I am not prepared to conclude, before hearing all the arguments, that any unconventional definition of a special function is necessarily a serious bug that needs to be fixed. It isn't possible to define special functions so that everyone's favorite identities will work out. Lots of things have changed in the past few decades, and the favored identities may have changed as well. The Mathematica definition of these functions does provide a more convenient form for some identities. Also, the Mathematica definition has a more convenient branch cut structure for certain purposes, such as numerical approximation on an an electronic computer, which probably wasn't a big concern when the original definitions were invented. I share your disappointment that the details of the reasons for departing from tradition in this example aren't more widely available. I think that this would make for an important contribution to the subject. Also, as has already been acknowledged, it would clearly be useful to correct the internal inconsistencies involving these functions, and to document any remaining unconventionalities more prominently. I hope that your request for free bug fixes and for information about these behaviors is already being addressed. Information about the design of Zeta and LerchPhi, for example, is available not only here in the Mathematica discussion group, but also in the Wolfram Research web site, or by contacting technical support. Dave Withoff Wolfram Research