Editing Elliptic integral (section)

==Complete elliptic integral of the third kind==
[[Image:Mplwp complete ellipticPi nfixed k.svg|thumb|300px|Plot of the complete elliptic integral of the third kind {{math|Π(''n'',''k'')}} with several fixed values of {{mvar|n}}]]
The '''complete elliptic integral of the third kind''' {{math|Π}} can be defined as

<math display="block">\Pi(n,k) = \int_0^\frac{\pi}{2} \frac{d\theta}{\left(1-n\sin^2\theta\right)\sqrt{1-k^2 \sin^2\theta}}.</math>

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the ''characteristic'' {{math|''n''}},
<math display="block">\Pi'(n,k) = \int_0^\frac{\pi}{2} \frac{d\theta}{\left(1+n\sin^2\theta\right)\sqrt{1-k^2 \sin^2\theta}}.</math>

Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean.{{sfn|Carlson|2010|loc=19.8}}

===Partial derivatives===
<math display="block">\begin{align}
\frac{\partial\Pi(n,k)}{\partial n} &= \frac{1}{2\left(k^2-n\right)(n-1)}\left(E(k)+\frac{1}{n}\left(k^2-n\right)K(k) + \frac{1}{n} \left(n^2-k^2\right)\Pi(n,k)\right) \\[8pt]

\frac{\partial\Pi(n,k)}{\partial k} &= \frac{k}{n-k^2}\left(\frac{E(k)}{k^2-1}+\Pi(n,k)\right)
\end{align}</math>