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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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1954

Authors and Affiliations

  • Paul F. Byrd
    • 1
  • Morris D. Friedman
    • 2
  1. 1.National Advisory Committee For Aeronautics (U.S.A.)Fisk UniversityPalo AltoUSA
  2. 2.National Advisory Committee For Aeronautics (U.S.A.)USA

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