Elliptic Integrals

Date: 12/13/2000 at 15:39:05
From: Lisa
Subject: Elliptic integrals

I am trying to find out exactly what elliptic integrals are and how to 
derive them. I've searched the Internet and the library but nothing 
helps me understand what they are.

Date: 12/13/2000 at 16:35:09
From: Doctor Rob
Subject: Re: Elliptic integrals

Thanks for writing to Ask Dr. Math, Lisa.

Integrals that can be transformed into one of the following three 
forms are elliptic integrals:

      t
     INT [1/sqrt([1-z^2]*[1-k^2*z^2]) dz]
      0

      t
     INT [sqrt([1-k^2*z^2]/[1-z^2]) dz]
      0

      t
     INT [1/[(1-a^2*z^2)*sqrt([1-z^2]*[1-k^2*z^2])] dz]
      0

These are called elliptic integrals of the first, second, and third 
kind, respectively.

More generally, you can define an elliptic integral to be an integral 
with respect to z of any rational function of z and sqrt(f(z)), where 
f(z) is a polynomial of degree 3 or 4. It can be shown that every 
integral like this can be reduced by a suitable change in variables to 
a sum of elementary functions and/or elliptic integrals of the first, 
second, and/or third kinds (above).

This may not be much help, but that's the definition.

The name was given them because the arc-length of an ellipse is given 
by an elliptic integral of the second kind, with k the eccentricity of 
the ellipse.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/