# CarlsonRM CarlsonRM[x,y,ρ]

gives Carlson's elliptic integral .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For non-negative arguments, .
• CarlsonRM[x,y,ρ] has a branch cut discontinuity at .
• CarlsonRM[x,y,ρ] is understood as a Cauchy principal value integral for ρ<0.
• For certain arguments, CarlsonRM automatically evaluates to exact values.
• CarlsonRM can be evaluated to arbitrary precision.
• CarlsonRM automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the function:

CarlsonRM is related to complete Legendre elliptic integral of the third kind:

## Scope(8)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate numerically to high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

### Specific Values(1)

Simple exact values are generated automatically:

### Derivatives(1)

Derivative of with respect to :

Derivative of with respect to :

## Applications(2)

Visualize the solid angle subtended by a circular disk:

Evaluate the solid angle:

Compare with the result of NIntegrate:

Visualize the intersection of a cylinder and a ball:

Volume of cylinder-ball intersection expressed in terms of Carlson integrals:

Compare with the result of Volume: