# Definition:Elliptic Integral

## Special Function

An elliptic integral is an integral in the form:

$\displaystyle \int_0^x R \left({t, \sqrt {P \left({t}\right)} }\right) \rd t$

where:

$P \left({t}\right)$ is a polynomial of degree $3$ or $4$
$R \left({t, \sqrt {P \left({t}\right)} }\right)$ is a rational function of $t$ and $\sqrt {P \left({t}\right)}$.

There exist some special cases:

## Elliptic Integral of the First Kind

### Incomplete Elliptic Integral of the First Kind

$\displaystyle F \left({k, \phi}\right) = \int \limits_0^\phi \frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi} }$

is the incomplete elliptic integral of the first kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$
$\phi$, defined on the interval $0 \le \phi \le \pi / 2$.

### Complete Elliptic Integral of the First Kind

$\displaystyle \map K k = \int \limits_0^{\pi / 2} \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$

is the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.

## Elliptic Integral of the Second Kind

### Incomplete Elliptic Integral of the Second Kind

$\displaystyle E \left({k, \phi}\right) = \int \limits_0^\phi \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi$

is the incomplete elliptic integral of the second kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$
$\phi$, defined on the interval $0 \le \phi \le \pi / 2$.

### Complete Elliptic Integral of the Second Kind

$\displaystyle E \left({k}\right) = \int \limits_0^{\pi / 2} \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi$

is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.

## Elliptic Integral of the Third Kind

### Incomplete Elliptic Integral of the Third Kind

$\displaystyle \Pi \left({k, n, \phi}\right) = \int \limits_0^\phi \frac {\mathrm d \phi} {\left({1 + n \sin^2 \phi}\right) \sqrt{1 - k^2 \sin^2 \phi} }$

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

$k$, defined on the interval $0 < k < 1$
$n \in \Z$
$\phi$, defined on the interval $0 \le \phi \le \pi / 2$.

### Complete Elliptic Integral of the Third Kind

$\displaystyle \Pi \left({k, n}\right) = \int \limits_0^{\pi / 2} \frac {\mathrm d \phi} {\left({1 + n \sin^2 \phi}\right) \sqrt{1 - k^2 \sin^2 \phi} }$

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

$k$, defined on the interval $0 < k < 1$
$n \in \Z$