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Evaluate Elliptic Integral of the first kind
Evaluate Elliptic Integral of the second kind
Evaluate Elliptic Integral of the third kind
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Carlson's Elliptic Integrals

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019

This tool evaluates the Carlson's symmetric form of elliptic integrals: RF(x,y,z), RC(x,y), RD(x,y,z) and RJ(x,y,z,p). Arguments x, y, z should be generally non-negative, but more restrictions apply. Argument p should be non-zero. Enter the arguments below.

x =
y =
z =
p =
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Result:
RF (x,y,z) =
RC (x,y) =
RD (x,y,z) =
RJ (x,y,z,p) =
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About Carlson's symmetric form of Elliptic Integrals

Definitions

The Carlson's symmetric form of the elliptic integral of the first kind is defined as:

R_F(x,y,z) = \frac{1}{2}\int_{0}^{\infty}\frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}

The Carlson's degenerate elliptic integral of the first kind, with y=z, is defined as:

R_C(x,y) = R_F(x,y,y) = \frac{1}{2}\int_{0}^{\infty}\frac{dt}{(t+y)\sqrt{(t+x)}}

The Carlson's symmetric form of the elliptic integral of the second kind is defined as:

R_D(x,y,z) = \frac{3}{2}\int_{0}^{\infty}\frac{dt}{(t+z)\sqrt{(t+x)(t+y)(t+z)}}

The Carlson's symmetric form of the elliptic integral of the third kind is defined as:

R_J(x,y,z,p) = \frac{3}{2}\int_{0}^{\infty}\frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)}}

See also
Evaluate Elliptic Integral of the first kind
Evaluate Elliptic Integral of the second kind
Evaluate Elliptic Integral of the third kind
All evaluation tools