Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 34

Incomplete Elliptic Integral of the First Kind
$34.1 \quad \displaystyle u = \map F {k, \phi} = \int \limits_0^\phi \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} } = \int \limits_0^x \frac {\d v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2}} }$

where $\phi = \operatorname {am} u$ is called the amplitude of $u$ and $x = \sin \phi$, where here and below $0 < k < 1$.

Complete Elliptic Integral of the First Kind
$34.2 \quad \displaystyle K = \map F {k, \pi / 2} = \int \limits_0^{\pi / 2} \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} } = \int \limits_0^1 \frac {\d v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

Incomplete Elliptic Integral of the Second Kind
$34.3 \quad \displaystyle \map E {k, \phi} = \int \limits_0^\phi \sqrt {1 - k^2 \sin^2 \phi} \rd \phi = \int \limits_0^x \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2} } \rd v$

Complete Elliptic Integral of the Second Kind
$34.4 \quad \displaystyle E = \map E {k, \pi / 2} = \int \limits_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \rd \phi = \int \limits_0^1 \dfrac {\sqrt{1 - k^2 v^2} } {\sqrt {1 - v^2} } \rd v$

Incomplete Elliptic Integral of the Third Kind
$34.5 \quad \displaystyle \map \Pi {k, n, \phi} = \int \limits_0^\phi \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt {1 - k^2 \sin^2 \phi} } = \int \limits_0^x \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

Complete Elliptic Integral of the Third Kind
$34.6 \quad \displaystyle \map \Pi {k, n, \pi / 2} = \int \limits_0^{\pi / 2} \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt{1 - k^2 \sin^2 \phi} } = \int \limits_0^1 \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$