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Difference between revisions of "Modulus of an elliptic integral"

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The parameter $k$ which enters into the expression of the [[Elliptic integral|elliptic integral]] in Legendre normal form. For example, in the incomplete elliptic integral of the first kind,
 
  
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$$F(\phi,k)=\int_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}}.\tag{*}$$
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{{MSC|33E05}}
  
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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 sharp 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|Jacobi elliptic functions]], which arise from the inversion of elliptic integrals of the form \ref{*}.
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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,
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$$F(\phi,k)=\int_0^\phi\frac{dt}{\sqrt{1-k^2\sin^2t}}.\label{*}\tag{*}$$
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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{*}.
  
  

Latest revision as of 15:40, 14 February 2020


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,

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

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)
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=32775
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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