`ellipticK`

Complete elliptic integral of the first kind.

Category: Mathematics > Mathematical Functions

Syntax

K = ellipticK(m)

Evaluates the integral from 0 to pi/2 of 1/sqrt(1 - m*sin(t)^2).

Description

K(m) is a quarter period of the Jacobi elliptic functions. It also gives the exact period of a pendulum swinging through a large angle theta0: T = 4*sqrt(L/g)*ellipticK(sin(theta0/2)^2), which reduces to 2*pi*sqrt(L/g) for small swings. It is related to the arithmetic-geometric mean by K = pi/(2*agm(1, sqrt(1 - m))).

Input arguments

m: Parameter (m = k^2), m <= 1 for real values.

Output arguments

K: Array of the same size as m. ellipticK(0) is pi/2 and ellipticK(1) is Inf.

Example

Special values and a pendulum factor

ellipticK(0)
ellipticK(sym(1))
ellipticK([0.1 0.5 0.9])
ellipticK(sin(pi/3)^2)/(pi/2)

Output

ans = 1.5708
ans = Inf
ans = 1×3

    1.6124    1.8541    2.5781
ans = 1.3729

See also

ellipticE ellipticF ellipticCK ellipticNome ellipke