Incomplete Elliptic Integral of the Second Kind E(φ,k)

Compute the incomplete elliptic integral of the second kind E(φ,k) from amplitude φ and modulus k using Carlson symmetric forms, with the complete integral E(k) and an integrand graph.

Input

Enter the amplitude φ (radians) and modulus k (between −1 and 1) to compute the incomplete elliptic integral of the second kind E(φ,k) using Carlson symmetric forms.

Amplitude φ (radians)

Upper limit of integration. Any real number (e.g. π/4 ≈ 0.7854)

Modulus k

A value from −1 to 1. The parameter is m = k squared

Result

Value of E(φ=0.785398, k=0.5)

0.7671959857

E(φ,k) / φ (average)

0.9768242676

Complete integral E(k=0.5)

1.4674622093

Parameter m = k squared

0.25

Integrand sqrt(1 − k²sin²θ) from 0 to φ

How it works

The incomplete elliptic integral of the second kind is defined as E(φ,k)=∫_0^φ sqrt(1−k^2 sin^2 θ) dθ, where φ is the amplitude and k is the modulus.
This tool takes the modulus k as input. Do not confuse it with the parameter m=k^2 (some references write E(φ|m) using m as the argument).
The computation uses Carlson symmetric integrals R_F and R_D: E=sin(φ)·R_F(cos^2 φ, 1−k^2 sin^2 φ, 1) − (k^2 sin^3 φ /3)·R_D(cos^2 φ, 1−k^2 sin^2 φ, 1).
To keep the integrand real, |k| must be at most 1. Any real φ is accepted, evaluated via the quasi-periodicity E(φ+π)=E(φ)+2E(k).
The Stat shows E(φ,k)/φ, the average value of the integrand. When φ=π/2 the result equals the complete elliptic integral E(k).

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