Modulus of an elliptic integral

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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,


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{*}.



[a1] F. Bowman, "Introduction to elliptic functions with applications" , Dover, reprint (1961)
How to Cite This Entry:
Modulus of an elliptic integral. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article