Method on @sym: ellipticE (m) ¶

Method on @sym: ellipticE (phi, m) ¶

Complete and incomplete elliptic integrals of the second kind.

The incomplete elliptic integral of the second kind with amplitude phi and parameter m is given by:

syms phi m
ellipticE (phi, m)
  ⇒ ans = (sym) E(φ│m)

rewrite (ans, 'Integral')
  ⇒ ans = (sym)
      φ
      ⌠
      ⎮    _________________
      ⎮   ╱        2
      ⎮ ╲╱  - m⋅sin (t) + 1  dt
      ⌡
      0

The complete elliptic integral of the second kind with parameter m is given by:

ellipticE (m)
  ⇒ ans = (sym) E(m)

rewrite (ans, 'Integral')
  ⇒ ans = (sym)
      π
      ─
      2
      ⌠
      ⎮    _________________
      ⎮   ╱        2
      ⎮ ╲╱  - m⋅sin (t) + 1  dt
      ⌡
      0

Examples:

vpa (ellipticE (sym (1), sym (1)/10))
  ⇒ (sym) 0.98620694978157550636951680164874

vpa (ellipticE (sym (-pi)/4))
  ⇒ (sym) 1.8443492468732292114663773247580

There are other conventions for the inputs of elliptic integrals, see ‘@sym/ellipticF’.

See also: @@sym/ellipke, @@sym/ellipticK, @@sym/ellipticPi.

Package: symbolic