# CarlsonRC

CarlsonRC[x,y]

gives the Carlson's elliptic integral .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For and , .
• CarlsonRC[x,y] has a branch cut discontinuity at .
• CarlsonRC[x,y] is real valued for and , and is interpreted as a Cauchy principal value integral for .
• For certain arguments, CarlsonRC automatically evaluates to exact values.
• CarlsonRC can be evaluated to arbitrary numerical precision.
• CarlsonRC automatically threads over lists.
• CarlsonRC can be used with Interval and CenteredInterval objects. »

# Examples

open allclose all

## Basic Examples(3)

Evaluate numerically:

Plot the function:

CarlsonRC is related to the special case of for :

## Scope(9)

### Numerical Evaluation(6)

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:

CarlsonRC can be used with Interval and CenteredInterval objects:

### Specific Values(1)

Simple exact values are generated automatically:

### Derivatives(1)

Derivative of with respect to :

Derivative of with respect to :

## Applications(3)

Use CarlsonRC to provide upper and lower bounds for CarlsonRF[x,y,z]:

CarlsonRC is useful for compactly expressing the change of parameter relations for EllipticPi:

Use CarlsonRC to express the change of parameter relation for CarlsonRJ:

## Properties & Relations(2)

For , can be expressed in terms of ArcCos: is interpreted as a Cauchy principal value if lies on the negative real axis:

Compare with the equivalent expression in terms of positive arguments:

## Possible Issues(1)

Generically, ; however, due to differences in the analytic structures of the two functions, the evaluation is restricted to numeric values of that do not lie on the negative real axis: