CarlsonRC

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 , being understood 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.

Examples

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Basic Examples (3)

Evaluate numerically:

Plot the function:

CarlsonRC is related to the special case of for :

Scope (8)

Numerical Evaluation (5)

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 threads elementwise over lists:

Specific Values (1)

Simple exact values are generated automatically:

Derivatives (1)

Derivative of with respect to :

Derivative of with respect to :

Function Representations (1)

TraditionalForm formatting:

Applications (2)

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

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

Properties & Relations (2)

For , the can be expressed in terms of ArcCos:

is interpreted as a principal value for on the negative real axis:

Compare to the closed-form result in terms of positive arguments:

Possible Issues (1)

Generically, ; however, due to differences in analytic structures of the two functions, the evaluation is restricted to numeric values of that can be checked to lay off the negative real axis:

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