CarlsonRD

gives the Carlson's elliptic integral .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For non-negative arguments, $TemplateBox[{x, y, z}, CarlsonRD]⩵3/2int_0^infty(t+x)^(-1/2)(t+y)^(-1/2)(t+z)^(-3/2)dt$ .
CarlsonRD[x,y,z] has a branch cut discontinuity at .
For certain arguments, CarlsonRD automatically evaluates to exact values.
CarlsonRD can be evaluated to arbitrary precision.
CarlsonRD automatically threads over lists.
CarlsonRD can be used with Interval and CenteredInterval objects. »

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Evaluate numerically:

CarlsonRD is related to a combination of Legendre elliptic integrals restricted to :

Evaluate CarlsonRD numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRD threads elementwise over lists:

Simple exact values are generated automatically:

When one argument of CarlsonRD is zero, CarlsonRD can be expressed in terms of the complete elliptic integrals CarlsonRE and CarlsonRK:

Derivative of with respect to :

An equation relating CarlsonRD, CarlsonRF and CarlsonRG:

Some cyclic permutation identities for CarlsonRD:

CarlsonRD satisfies the Euler–Poisson partial differential equation:

CarlsonRD satisfies Euler's homogeneity relation:

Distance along a meridian of the Earth:

Compare with the result of GeoDistance:

Parametrization of a Mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):

CarlsonRD is symmetric with respect to its first two arguments:

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