Method on @sym: y = ellipticPi (nu, m) ¶

Method on @sym: y = ellipticPi (nu, phi, m) ¶

Complete and incomplete elliptic integrals of the third kind.

Incomplete elliptic integral of the third kind with characteristic nu, amplitude phi and parameter m:

syms nu phi m
ellipticPi (nu, phi, m)
  ⇒ (sym) Π(ν; φ│m)

rewrite (ans, 'Integral')         % doctest: +SKIP
  ⇒ ans = (sym)
      φ
      ⌠
      ⎮                   1
      ⎮ ────────────────────────────────────── dα
      ⎮    _________________
      ⎮   ╱        2         ⎛       2       ⎞
      ⎮ ╲╱  - m⋅sin (α) + 1 ⋅⎝- ν⋅sin (α) + 1⎠
      ⌡
      0

Complete elliptic integral of the third kind with characteristic nu and parameter m:

ellipticPi (nu, m)
  ⇒ ans = (sym) Π(ν│m)

rewrite (ans, 'Integral')         % doctest: +SKIP
  ⇒ ans = (sym)
      π
      ─
      2
      ⌠
      ⎮                   1
      ⎮ ────────────────────────────────────── dα
      ⎮    _________________
      ⎮   ╱        2         ⎛       2       ⎞
      ⎮ ╲╱  - m⋅sin (α) + 1 ⋅⎝- ν⋅sin (α) + 1⎠
      ⌡
      0

Examples:

vpa (ellipticPi (sym (1), sym (1)/10, sym (1)/2))
  ⇒ (sym) 0.10041852861527457424263837477419

vpa (ellipticPi (sym (pi)/4, sym (pi)/8))
  ⇒ (sym) 4.0068172051461721205075153294257

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

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

Package: symbolic