Modulus of an elliptic integral

2020 Mathematics Subject Classification: Primary: 33E05 [MSN][ZBL]

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,

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

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 \ref{*}.

References

[a1]	F. Bowman, "Introduction to elliptic functions with applications" , Dover, reprint (1961)