Complete Elliptic Integral K(k) and E(k) Calculator

Enter the modulus k to compute the first kind complete elliptic integral K(k) and the second kind E(k) with high precision using the arithmetic geometric mean (AGM) method.

Input

Enter the modulus k (from 0 up to but not including 1) to compute the first kind complete elliptic integral K(k) and the second kind E(k) by the arithmetic geometric mean (AGM) method.

Modulus k

A real number from 0 up to but not including 1 (k convention; differs from the parameter m equal to k squared)

Result

First kind complete elliptic integral K(k) at k = 0.5

1.6857503548

Second kind E(k) at k = 0.5

1.4674622093

Modulus k

0.5

Parameter m equal to k squared

0.25

K of complementary modulus K(k'')

2.1565156475

Graph of K(k) and E(k)

K(k) first kind

E(k) second kind

How it works

Computes the first kind complete elliptic integral K(k) and the second kind E(k) using the arithmetic geometric mean (AGM) method.
The input is the modulus k. Values from 0 up to but not including 1 are accepted. This tool uses the k convention, so be careful not to confuse it with the parameter m equal to k squared.
The AGM iteration starts from a0=1 and b0 equal to the square root of (1 minus k squared); the converged mean M gives K(k) = pi / (2M).
E(k) is obtained from the sum of squares of the differences c_n produced during the AGM iteration as E(k) = K(k) times (1 minus the sum of 2^(n-1) times c_n squared).
The K value for the complementary modulus, the square root of (1 minus k squared), is also shown.
As k approaches 1 the value K(k) diverges logarithmically. The graph clips the vertical axis at a fixed value for readability.

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