CarlsonRG

gives the Carlson's elliptic integral .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For non-negative arguments, $TemplateBox[{x, y, z}, CarlsonRG]=1/4int_0^infty(t+x)^(-1/2)(t+y)^(-1/2)(t+z)^(-1/2)((x t)/(t+x)+(y t)/(t+y)+(z t)/(t+z))dt$ .
CarlsonRG[x,y,z] has a branch cut discontinuity at .
For certain arguments, CarlsonRG automatically evaluates to exact values.
CarlsonRG can be evaluated to arbitrary precision.
CarlsonRG automatically threads over lists.
CarlsonRG can be used with Interval and CenteredInterval objects. »

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Evaluate numerically:

Plot over a range of arguments:

CarlsonRG is related to the Legendre elliptic integral of the second kind for :

Evaluate numerically:

Evaluate at high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRG threads elementwise over lists:

Simple exact results are generated automatically:

When one argument of CarlsonRG is zero, CarlsonRG reduces to the complete elliptic integral CarlsonRE:

When two of the arguments of CarlsonRG are identical and do not lie on the negative real axis, CarlsonRG can be expressed in terms of CarlsonRC:

When all arguments of CarlsonRG are identical and do not lie on the negative real axis, CarlsonRG reduces to an elementary function:

Derivative of with respect to :

An equation relating CarlsonRG, CarlsonRF and CarlsonRD:

CarlsonRG satisfies the Euler–Poisson partial differential equation:

CarlsonRG satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRG:

Calculate the surface area of a triaxial ellipsoid:

The area of an ellipsoid with semi‐axes 3, 2, 1:

Use RegionMeasure to calculate the surface area of the ellipsoid:

Expectation value of the square root of a quadratic form over a normal distribution:

Compare with the closed-form result in terms of CarlsonRG:

CarlsonRG is invariant under a permutation of its arguments:

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