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Re: Re: Mathematica 3.0.0 bug in LerchPhi function

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

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

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