Incomplete Elliptic Integral of the First Kind F(φ,k)

Compute the incomplete elliptic integral of the first kind F(φ,k) from amplitude φ and modulus k using the Carlson symmetric form, with the K(k) ratio.

Input

Enter the amplitude φ and modulus k to compute the incomplete elliptic integral of the first kind F(φ,k) via the Carlson symmetric form.

Amplitude φ

Angle at the upper limit of the integral

Angle unit

Modulus k

A value from −1 to 1 (enter k, not m where m=k²)

Result

F(φ=45degrees, k=0.5)

0.8043661012

Ratio K(k)/F(φ,k)

2.095750122

Complete integral K(k)

1.6857503548

Integrand 1/√(1−k²sin²θ)

Calculation details

Amplitude φ (radians)	0.7853981634
Modulus k	0.5
Parameter m = k²	0.25
F(φ, k)	0.8043661012
Complete integral K(k)	1.6857503548

How it works

Computes the incomplete elliptic integral of the first kind F(φ,k)=∫_0^φ dθ/√(1−k²sin²θ), where φ is the amplitude and k is the modulus.
The modulus convention uses k. In the parameter m convention, m=k², so enter k here rather than m.
The calculation uses the Carlson symmetric form R_F, evaluating F(φ,k)=sin(φ)·R_F(cos²φ, 1−k²sin²φ, 1).
R_F is found by repeated duplication until the arguments converge, then a Taylor expansion about their mean, which is numerically stable.
For φ beyond π/2 the periodicity F(φ+nπ,k)=2nK(k)+F(φ,k) is used to reduce to the principal interval.
The Stat shows the complete elliptic integral K(k) and the ratio K(k)/F(φ,k). When φ equals π/2 this ratio is 1.
Enter the modulus k in the range −1 to 1. Outside this range the integral is not real valued.
The angle unit can be degrees or radians. Degrees are converted internally to radians for the computation.

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