`ellipticE`

Complete and incomplete elliptic integrals of the second kind.

Category: Mathematics > Mathematical Functions

Syntax

E = ellipticE(m)

Complete integral of the second kind with parameter m.

E = ellipticE(phi, m)

Incomplete integral with amplitude phi in radians.

Description

The incomplete integral is integral from 0 to phi of sqrt(1 - m*sin(t)^2) dt and the complete one takes phi = pi/2. The complete integral gives the perimeter of an ellipse: with semi-axes a >= b, the perimeter is 4*a*ellipticE(1 - b^2/a^2). Note that MATLAB uses the parameter m, not the modulus k.

Input arguments

m: Parameter, the square of the modulus: m = k^2.
phi: Amplitude angle in radians.

Output arguments

E: Array of the same size as the inputs.

Example

Ellipse perimeter

a = 2; b = 1;
P = 4*a*ellipticE(1 - b^2/a^2)
ellipticE(pi/4, 0.5)

Output

P = 9.6884
ans = 0.7482

ellipticE