Modulus of an elliptic integral
This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.
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{*}.
Comments
References
| [a1] | F. Bowman, "Introduction to elliptic functions with applications" , Dover, reprint (1961) |
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