IncompleteEllipticPi

The incomplete Legendre elliptic integral of the third kind, $\Pi(n;\varphi,m) = \int_0^\varphi \dfrac{d\theta}{(1-n\sin^2\theta)\sqrt{1-m\sin^2\theta}}$, in Wolfram/mpmath's $(n,\varphi,m)$ order with $m=k^2$.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimIncompleteEllipticPi✓↗Fungrim entry255d81✓↗5f84d9✓↗8f4e31↗Wolfram LanguageEllipticPi!↗

IncompleteEllipticPi(n, phi, m)the incomplete elliptic integral of the third kind, characteristic n, amplitude φ, parameter m.

Typed signatures

Type	Package	Applies
(number, number, number) -> number	analytic	always

Domain: Special functions

Details

• $m = k^2$, the same convention EllipticE / EllipticF use — not the elliptic modulus k itself.
• $\varphi = \pi/2$ is the complete integral: $\Pi(n;\pi/2,m) = $ EllipticPi$(n,m)$.
• Built from Carlson's symmetric $R_F$/$R_J$ (DLMF 19.25.14) rather than delegating to compute-engine's native three-argument EllipticPi, which returns NaN for some complex φ inside its own stated domain.
• Quasi-periodic: $\Pi(n;\varphi+k\pi,m) = 2k\,\Pi(n,m) + \Pi(n;\varphi,m)$ for integer k, so any φ reduces to $[-\pi/2,\pi/2]$ before the Carlson evaluation.
• Numeric only — a symbolic or exact argument stays unevaluated; a floating-point argument (or N()) evaluates directly, same as EllipticE and EllipticF.

Examples

11 more kept as data, checked against the oracles

Basic #

Π(n; 0, m) = 0

live ↗

In$\mathrm{IncompleteEllipticPi}(0.5, 0, 0.3)$Out$0$

A floating-point argument evaluates directly to a decimal

live ↗

In$N(\mathrm{IncompleteEllipticPi}(0.5, 0.4, 0.3))$Out$0.414\,151\,736\,824\,476\,76$

Known value · mpmath 1.3.0 ellippi

Scope #

A complex amplitude φ, where native EllipticPi returns NaN even though φ is inside its own stated domain

live ↗

In$N(\mathrm{IncompleteEllipticPi}(0.2, 1.2+0.5i, 0.3))$Out$1.321\,415\,376\,117\,118+0.718\,665\,718\,875\,180\,6i$

Known value · mpmath 1.3.0 ellippi

All three arguments complex — matches mpmath's ellippi

live ↗

In$N(\mathrm{IncompleteEllipticPi}(1.17+0.45i, 1.17+0.45i, 1.17+0.45i))$Out$0.343\,014\,061\,827\,736\,86+1.078\,685\,669\,354\,995\,8i$

Characteristic $n > 1$

live ↗

In$N(\mathrm{IncompleteEllipticPi}(0.3, 1.1, 0.4))$Out$1.323\,800\,815\,047\,493\,6$

Properties #

n = 0 reduces to the incomplete elliptic integral of the first kind, IncompleteEllipticF(φ, m)

live ↗

In$N(\mathrm{IncompleteEllipticPi}(0, 0.4, 0.3))$Out$0.403\,164\,991\,947\,139\,34$

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

live ↗

In$\mathrm{IncompleteEllipticPi}(n, ϕ, m)$Out$\mathrm{IncompleteEllipticPi}(n, ϕ, m)$

arccsc(0) is ComplexInfinity; stays symbolic here — Wolfram's own EllipticPi[0, ArcCsc[0], 0] answers Indeterminate (shown as NaN).

live ↗

In$\mathrm{IncompleteEllipticPi}(0, \mathrm{arccsc}(0), 0)$Out$\mathrm{IncompleteEllipticPi}(0, \tilde∞, 0)$

Same as above, shifted by π: arccsc(0) still dominates and the result stays symbolic.

live ↗

In$\mathrm{IncompleteEllipticPi}(0, π+\mathrm{arccsc}(0), 0)$Out$\mathrm{IncompleteEllipticPi}(0, \tilde∞, 0)$

About

Examples	20
Checked against	Wolfram Language 18 agree2 differ
Crosswalk	FungrimFungrim entry ×3Wolfram Language

See also: EllipticPi, EllipticE, EllipticF