Method on @sym: ellipticK (m) ¶

Complete elliptic integral of the first kind.

The complete elliptic integral of the first kind with parameter m is defined by:

syms m
ellipticK (m)
  ⇒ ans = (sym) K(m)

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

Examples:

diff (ellipticK (m), m)
  ⇒ (sym)
      -(1 - m)⋅K(m) + E(m)
      ────────────────────
          2⋅m⋅(1 - m)

vpa (ellipticK (sym (pi)/4))
  ⇒ (sym) 2.2252536839853959577044373301346

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

See also: @@sym/ellipke, @@sym/ellipticF, @@sym/ellipticE, @@sym/ellipticPi.

Package: symbolic