Elliptic IntegralsDate: 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/ |
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