From: Dave Rusin <rusin@math.niu.edu>
Subject: Re: are these integrals known.
Date: Fri, 5 Mar 1999 01:25:50 -0600 (CST)
Newsgroups: [missing]
To: verberkm@wins.uva.nl

Dear Mr Verberkmoes:

On 19 Nov 1998 you submitted a post to sci.math.research about some integrals
from statistical mechanics:

>Let b be a complex number and c its complex conjugate.  Define
>
>          (z-1/z)-(b-1/b)  1/6
>t(z) := ( --------------- )
>          (z-1/z)-(c-1/c)
>
>The integrals I need are:
>
>        b
>        /  t(z)+1/t(z)
>(A)     \  ----------- dz
>        /       z
>        c
>

[etc.]

You wanted to know if these integrals have a "closed form", and I
explained that they couldn't, being related to functions which live on
a Riemann surface of genus  5  while all the standard functions are
associated to surfaces of genus 0 and 1.

That response was correct, I still think, but perhaps irrelevant, since
you were asking for _definite_ integrals, not antiderivatives. I think I
should amend my answer to say, there is _probably_ no closed-form expression
for these integrals. I have recently been alerted to the fact that the
definite integrals associated to periods on Riemann surfaces may in fact
have an integral which can be expressed more easily than should be expected
from the general theory. Moreover, it is possible in some of those cases
actually to find expressions for the integrals. A case in point is
the integral from 0 to 1 of  sqrt(x(1-x)(1+x^2)): the antiderivative is a
function associated to a curve of genus  2, and so cannot be expressed
via the trig and elliptic functions, for example; yet the _definite integral
from 0 to 1_ does admit a complicated expansion in terms of those functions.
This topic has just arisen on  sci.math.symbolic, should you wish to follow
this discussion. I applied the methods of that discussion to your problem;
after much deliberation, Maple conceded it couldn't find antiderivatives
of the type it needed (without however claiming that it had proved no such
antiderivatives exist).

I hope I have not led you too far astray with my incomplete answer! If you
think further information along these lines is important for your work,
I'd be happy to post your requests to sci.math.research.

dave

==============================================================================
For the corresponding indefinite integrals, see math-atlas.org/98/cmplx_int .
==============================================================================

Date: Sun, 04 Nov 2001 23:18:28 +0100
To: rusin@math.niu.edu
From: Alain Verberkmoes <verberkm@xs4all.nl>
Subject: acknowledgement in paper published

[deletia -- djr]

The integrals came up in our study of a solvable lattice
model in statistical mechanics.  We have published our
findings in two papers:
- A. Verberkmoes and B. Nienhuis,
  Phys. Rev. Lett. 83 (1999) 3986.
- A. Verberkmoes and B. Nienhuis,
  Phys. Rev. E 63 (2001) 066122.
It is my pleasure to inform you that your help has been
acknowledged (in the latter paper).

Yours,
Alain Verberkmoes.

PS  In case you're interested, a preliminary version of each
article is available on-line:
http://xxx.lanl.gov/abs/cond-mat/9904343
http://xxx.lanl.gov/abs/cond-mat/9909068