Elliptic Integral - Incomplete Elliptic Integral of The Third Kind

The incomplete elliptic integral of the third kind Π is

\Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi \frac{1}{1-n\sin^2 \theta} \frac {d\theta}{\sqrt{1-(\sin\theta\sin \alpha)^2}}, or
\Pi(n ; \varphi \,|\,m) = \int_{0}^{\sin \varphi} \frac{1}{1-nt^2} \frac{dt}{\sqrt{(1-m t^2)(1-t^2) }}.

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value is infinite, for any m.

A relation with the Jacobian elliptic functions is

The meridian arc length from the equator to latitude is also related to a special case of Π:

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