info prev up next book cdrom email home

Elliptic Integral of the Third Kind

Let $0 < k^2 < 1$. The incomplete elliptic integral of the third kind is then defined as

$\displaystyle \Pi(n;\phi,k)$ $\textstyle =$ $\displaystyle \int_0^\phi {d\theta \over (1-n\sin^2\theta)\sqrt{1-k^2 \sin^2\theta}}$ (1)
  $\textstyle =$ $\displaystyle \int_0^{\sin \phi} {dt \over (1-nt^2)\sqrt{(1-t^2)(1-k^2t^2)}},$ (2)

where $n$ is a constant known as the Characteristic.


The complete elliptic integral of the second kind

\Pi(n\vert m)=\Pi(n; {\textstyle{1\over 2}}\pi \vert m)
\end{displaymath} (3)

is illustrated above.

See also Elliptic Integral of the First Kind, Elliptic Integral of the Second Kind, Elliptic Integral Singular Value


Abramowitz, M. and Stegun, C. A. (Eds.). ``Elliptic Integrals'' and ``Elliptic Integrals of the Third Kind.'' Ch. 17 and §17.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.

© 1996-9 Eric W. Weisstein