Method on @sym: ellipticF (phi, m) ¶

Incomplete elliptic integral of the first kind.

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

syms phi m
ellipticF (phi, m)
  ⇒ ans = (sym) F(φ│m)

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

Example:

vpa (ellipticF (sym (1), sym (-1)))
  ⇒ (sym) 0.89639378946289458637047451642060

For the complete elliptic integral (of the first kind), see ‘@sym/ellipticK’.

Note: this function (and other elliptic integrals in the Symbolic package) follow the Abramowitz and Stegun convention for the “parameter” m. Other sources and software may use different conventions, such as the “elliptic modulus” k or the “modular angle” α, related by m = k^2 = sin^2(α). They may define these functions in terms of the sine of the amplitude sin(phi). For example, Maple uses the elliptic modulus and the sine of the amplitude.

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

Package: symbolic