Complete Elliptic Integral of the Third Kind Π(n,k)

Enter the characteristic n and modulus k to compute the complete elliptic integral of the third kind Π(n,k) with high precision. Also shows K(k) and a curve over the modulus.

Input

Enter the characteristic n and the modulus k to compute the complete elliptic integral of the third kind Pi(n,k). The modulus is handled internally as m=k squared.

Characteristic n

The value entering (1 minus n sin^2 theta) in the integrand. A real principal value requires n less than 1.

Modulus k

The modulus entering sqrt(1 minus k^2 sin^2 theta). Its absolute value must be less than 1 for convergence.

Result

Value of Pi(n=0.5, k=0.5)

2.4136715042

Characteristic n

0.5

Modulus k

0.5

Squared modulus m = k^2

0.25

First kind K(k)

1.6857503548

Pi(n,k) versus the modulus k (n fixed)

How it works

The complete elliptic integral of the third kind is defined as Pi(n,k)=integral from 0 to pi/2 of dtheta/((1 minus n sin^2 theta) times sqrt(1 minus k^2 sin^2 theta)), where theta is the variable of integration, n is the characteristic and k is the modulus.
Modulus convention: this tool takes the modulus k as input and uses the parameter m=k^2 internally. Some references write Pi(n,m) with the squared modulus m as the second argument, so check the convention when comparing reference values.
The computation uses an independent implementation of Carlson symmetric integrals via the identities K(k)=R_F(0,1 minus m,1) and Pi(n,k)=K(k)+(n/3) times R_J(0,1 minus m,1,1 minus n).
Convergence requires the modulus to satisfy abs(k) less than 1 (m less than 1). For characteristic n greater than or equal to 1 the integrand has a singularity and is not a real principal value, so this tool treats it as out of range.
When the characteristic n equals 0, Pi(0,k) reduces to the first-kind complete elliptic integral K(k). The chart fixes the characteristic n and shows how Pi(n,k) varies as the modulus k increases from 0.

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