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.