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') % doctest: +SKIP ⇒ ans = (sym) π ─ 2 ⌠ ⎮ 1 ⎮ ────────────────── dα ⎮ _______________ ⎮ ╱ 2 ⎮ ╲╱ 1 - m⋅sin (α) ⌡ 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