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 , 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 threads elementwise over lists:

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 :

Function Representations (1)

TraditionalForm formatting:

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:

Top

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