# 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,

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

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{*}.

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#### 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=44714