Elliptic Functions

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Originally motivated by the computation of the arc length of an ellipse, Jacob Jacobi introduced the theory of Jacobi elliptic functions in the book Fundamenta nova theoriae functionum ellipticarum (New foundations of the theory of the elliptic functions) in 1829. Jacobi elliptic functions are doubly periodic (in the real and imaginary directions) and meromorphic (analytic with the possible exception of isolated poles).

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Elliptic functions appear in problems like the planar pendulum, motion in a cubic or quartic potential, the force-free asymmetric top and the heavy symmetric top with one fixed point, wave solutions in the KdV equation, the translational partition function for an ideal gas, geodesics in general relativity, and in cosmological models.

The plots show elliptic functions with arguments of the form .

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Contributed by: Enrique Zeleny (November 2014)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Elliptic functions originally arose from the inversion of the integral

,

known as the incomplete elliptic integral of the first kind, where the angle is the amplitude and is the modulus. Then define

,

,

.

The reciprocals of the functions , , are named by switching the order of the two letters to , , ; the ratios of the functions , , are named by combining the first letters of the functions in the denominator to , , and , , .

Also included in this Demonstration are other types of elliptic functions: the Weierstrass elliptic function , the Dixon elliptic functions and , and the Gauss lemniscate functions and . The Dixon functions and the Weierstrass and the square of its derivative show the symmetries of some of the wallpaper groups. The Dixon functions are defined by

,

.

The Gauss lemniscate functions are defined by

,

.

References

[1] I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., San Diego: Academic Press, 2000.

[2] L. M. Milne-Thomson, "Jacobi Elliptic Functions and Theta Functions," in Handbook of Mathematical Functions, (M. Abramowitz and I. A. Stegun, eds.), New York: Dover, 1965.

[3] W. P. Reinhardt and P. L. Walker, "Chapter 22: Jacobian Elliptic Functions," NIST Digital Library of Mathematical Functions, Version 1.0.9; Release date 2014-08-29. dlmf.nist.gov/22.

[4] Souichiro-Ikebe. "Elliptic Functions." Graphics Library of Special Functions (in Japanese). (Dec 4, 2015) http://math-functions-1.watson.jp/sub1_spec_090.html.

[5] A. C. Dixon, The Elementary Properties of the Elliptic Functions, with Examples, London: Macmillan, 1894. archive.org/details/117736039.

[6] A. G. Greenhill, The Applications of Elliptic Functions, London, New York: Macmillan, 1892.

[7] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., Cambridge: Cambridge University Press, 1937.

[8] V. G. Tkachev. "Elliptic Functions: Introduction Course." (Nov 7, 2014) www.mai.liu.se/~vlatk48/papers/lect2-agm.pdf.

[9] A. J. Brizard, "A Primer on Elliptic Functions with Applications in Classical Mechanics," arxiv.org/pdf/0711.4064v1.pdf.

[10] Deoxygerbe. "Elliptic Functions on the 17 Wallpaper Groups." Mathematics StackExchange. (May 3, 2011) math.stackexchange.com/questions/36737/elliptic-functions-on-the-17-wallpaper-groups?rq=1.



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