gives the complete elliptic integral of the third kind Π(nm).
gives the incomplete elliptic integral Π(n;ϕm).
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For real , , and , where the principal value integral is understood for .
- EllipticPi[n,m] has branch cut discontinuities at and at .
- EllipticPi[n,ϕ,m] has branch cut discontinuities at , at and at .
- For certain special arguments, EllipticPi automatically evaluates to exact values.
- EllipticPi can be evaluated to arbitrary numerical precision.
- EllipticPi automatically threads over lists.
- EllipticPi can be used with Interval and CenteredInterval objects. »
Examplesopen allclose all
Basic Examples (6)
Series expansion at Infinity:
Numerical Evaluation (6)
Evaluate EllipticPi efficiently at high precision:
EllipticPi threads elementwise over lists:
Specific Values (3)
Function Properties (9)
EllipticPi is not an analytic function:
EllipticPi is not a meromorphic function:
Series Expansions (3)
The action can be expressed using EllipticPi (for brevity, occurring roots are abbreviated):
Properties & Relations (4)
Wolfram Research (1988), EllipticPi, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticPi.html (updated 2022).
Wolfram Language. 1988. "EllipticPi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticPi.html.
Wolfram Language. (1988). EllipticPi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EllipticPi.html