Definition:Elliptic Integral of the Third Kind/Complete

Special Function

Definition 1

$\ds \map \Pi {k, n} = \int \limits_0^{\pi / 2} \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt {1 - k^2 \sin^2 \phi} }$

is the complete elliptic integral of the third kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$

$n \in \Z$

Definition 2

$\ds \map \Pi {k, n} = \int \limits_0^1 \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

is the complete elliptic integral of the third kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$

$n \in \Z$

Also see

• Equivalence of Definitions of Complete Elliptic Integral of the Third Kind

• Definition:Incomplete Elliptic Integral of the Third Kind

• Definition:Elliptic Integral of the First Kind:
  • Definition:Incomplete Elliptic Integral of the First Kind
  • Definition:Complete Elliptic Integral of the First Kind

• Definition:Elliptic Integral of the Second Kind:
  • Definition:Incomplete Elliptic Integral of the Second Kind
  • Definition:Complete Elliptic Integral of the Second Kind

• Results about the complete elliptic integral of the third kind can be found here.