View source for Modulus of an elliptic integral

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{{MSC|33E05}}

The parameter $k$ which enters into the expression of the [[elliptic integral]] in Legendre normal form. For example, in the incomplete elliptic integral of the first kind,

$$F(\phi,k)=\int_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}}.\label{*}\tag{*}$$

The number $k^2$ is sometimes called the Legendre modulus, $k'=\sqrt{(1-k^2)}$ is called the complementary modulus. In applications the normal case $0<k<1$ usually holds; here the acute angle $\theta$ for which $\sin\theta=k$ is called the modular angle. The modulus $k$ also enters into the expression of the [[Jacobi elliptic functions]], which arise from the inversion of elliptic integrals of the form \eqref{*}.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bowman,   "Introduction to elliptic functions with applications" , Dover, reprint  (1961)</TD></TR></table>