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### 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/
```
Associated Topics:
College Calculus

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