From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: An Integration
Date: 23 Nov 1997 06:05:02 GMT

In article <34724731.19CD@rpi.edu>, Chunwei Wang  <wangc6@rpi.edu> wrote:
>Any hint for integrating this function to get an exact solution?
>
>	sqrt(A*x^3+B*x^2+C*x+D)
>
>	where A, B, C, and D are non-zero constants.

No closed form solution. You're asking us to integrate  y  over the
Riemann surface  y^2=Ax^3+Bx^2+Cx+D, but that surface is a torus, so the
antiderivative (actually a path integral) isn't quite well-defined;
paths which are nontrivial in the fundamental group will augment the
integral by a constant. Put another way, the inverse of the antiderivative
is a doubly-periodic function, the periods being just the integrals
around the closed loops of the Riemann surface. Well, do you know
any doubly-periodic functions which might even be candidates for  that
inverse function? I didn't think so.

(Yeah sure, there's the Weierstrass Pe function, but that's about as
useful here as the obvious response, "Define  F(x)  to be the antiderivative
of ...; then the integral in question is precisely  F.")

dave

==============================================================================
From: edgar@math.ohio-state.edu (G. A. Edgar)
Newsgroups: sci.math
Subject: Re: An Integration
Date: Sun, 23 Nov 1997 08:43:52 -0500

In article <34724731.19CD@rpi.edu>, Chunwei Wang <wangc6@rpi.edu> wrote:

> Any hint for integrating this function to get an exact solution?
> 
>         sqrt(A*x^3+B*x^2+C*x+D)
> 
>         where A, B, C, and D are non-zero constants.
> 
> Thanks!

This is an "elliptic integral".  Except for special choices of
the coefficients, it is not an elementary function.  But it
may be written in terms of the standard elliptic integrals
E, F, and Pi (or whatever they are called).
-- 
Gerald A. Edgar                   edgar@math.ohio-state.edu