CarlsonRJ

CarlsonRJ[x,y,z,ρ]

gives Carlson's elliptic integral .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For non-negative arguments, .
CarlsonRJ[x,y,z,ρ] has a branch cut discontinuity at .
CarlsonRJ[x,y,z,ρ] is understood as a Cauchy principal value integral for .
For certain arguments, CarlsonRJ automatically evaluates to exact values.
CarlsonRJ can be evaluated to arbitrary precision.
CarlsonRJ automatically threads over lists.
CarlsonRJ can be used with Interval and CenteredInterval objects. »

Examples

open allclose all

Basic Examples (3)

Evaluate numerically:

Visualize over a range of arguments:

CarlsonRJ is related to Legendre elliptic integral of the third kind for :

Scope (10)

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:

CarlsonRJ threads elementwise over lists:

CarlsonRJ can be used with Interval and CenteredInterval objects:

Specific Values (2)

Simple exact values are generated automatically:

The case of complete Carlson elliptic integral:

Derivatives (1)

Derivative of with respect to :

Derivative of with respect to :

Function Representations (1)

TraditionalForm formatting:

Applications (1)

Use CarlsonRJ to define a conformal map:

Visualize the image of lines of constant real and imaginary parts:

Top

Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how