Difference between revisions of "Modulus of an elliptic integral"
From Encyclopedia of Mathematics
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$$F(\phi,k)=\int_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}}.\tag{*}$$ | $$F(\phi,k)=\int_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}}.\tag{*}$$ | ||
| − | 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 | + | 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{*}. |
Revision as of 21:17, 13 November 2016
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{*}.
Comments
References
| [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: http://encyclopediaofmath.org/index.php?title=Modulus_of_an_elliptic_integral&oldid=34739
Modulus of an elliptic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modulus_of_an_elliptic_integral&oldid=34739
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article