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.

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 TemplateBox[{n, phi, m}, EllipticPi3] for :

Scope  (9)

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:

CarlsonRJ threads elementwise over lists:

Specific Values  (2)

Simple exact values are generated automatically:

The case of complete Carlson elliptic integral:

Derivatives  (1)

Derivative of TemplateBox[{x, y, z, w}, CarlsonRJ] with respect to :

Derivative of TemplateBox[{x, y, z, w}, CarlsonRJ] 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:

Wolfram Research (2021), CarlsonRJ, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRJ.html.

Text

Wolfram Research (2021), CarlsonRJ, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRJ.html.

CMS

Wolfram Language. 2021. "CarlsonRJ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CarlsonRJ.html.

APA

Wolfram Language. (2021). CarlsonRJ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarlsonRJ.html

BibTeX

@misc{reference.wolfram_2022_carlsonrj, author="Wolfram Research", title="{CarlsonRJ}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRJ.html}", note=[Accessed: 24-March-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_carlsonrj, organization={Wolfram Research}, title={CarlsonRJ}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRJ.html}, note=[Accessed: 24-March-2023 ]}