# Elliptic Integrals of the Third Kind

• Paul F. Byrd
• Morris D. Friedman
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (GL, volume 67)

## Abstract

The incomplete elliptic integral of the third kind in Legendre’s canonical form is defined by
$${400.01^*}\left\{ {\begin{array}{*{20}{l}} {\Pi \left( {\varphi ,{\alpha ^2},k} \right) \equiv \int\limits_0^y {\frac{{dt}}{{\left( {1 - {\alpha ^2}{t^2}} \right)\sqrt {\left( {1 - {t^2}} \right)\left( {1 - {k^2}{t^{^2}}} \right)} }}} = \int\limits_0^\varphi {\frac{{d\vartheta }}{{\left( {1 - {\alpha ^2}{{\sin }^2}\vartheta } \right)\sqrt {1 - {k^2}{{\sin }^2}\vartheta } }}} } \\ {\quad \quad \quad \quad \;\; = \int\limits_0^{{u_1}} {\frac{{du}}{{1 - {\alpha ^2}s{n^2}u}} \equiv \Pi \left( {{u_1},{\alpha ^2}} \right)} ,} \\ {\quad \quad \quad \quad \;\;\left[ {y = \sin \varphi = sn\;{u_1},\quad t = \sin \vartheta = sn\;u;\quad {\alpha ^2} \ne 1\;or\;{k^2}} \right].} \end{array}} \right.$$
. When ϕ=π/2, y-1, u 1=K, the integral is said to be complete.