# Definition:Elliptic Integral of the Third Kind/Incomplete

< Definition:Elliptic Integral of the Third Kind(Redirected from Definition:Incomplete Elliptic Integral of the Third Kind)

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## Special Function

### Definition 1

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

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 $0 \le \phi \le \pi / 2$.

### Definition 2

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

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

- $k$, defined on the interval $0 < k < 1$
- $n \in \Z$
- $x = \sin \phi$, where $\phi$ is defined on the interval $0 \le \phi \le \pi / 2$.

### Completion

- $\map \Pi {k, n, \dfrac \pi 2} = \map \Pi {k, n}$

where $\map \Pi {k, n}$ denotes the **complete** elliptic integral of the third kind.

## Also known as

Some sources omit the **incomplete** from the name, calling this merely the **elliptic integral of the third kind**.

## Also see

- Results about
**the incomplete elliptic integral of the third kind**can be found**here**.