# Elliptic Functions

• John Stillwell
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Preview

Elliptic functions, like many innovations in mathematics, arose as a way around an impasse. As we saw in Section 9.6, the search for closed-form solutions in integral calculus foundered on integrands such as $$1/\sqrt{1-x^4}$$, because no “known” function f(x) has derivative $$1/\sqrt{1-x^4}$$. Eventually, mathematicians accepted the fact that $$\int_0^x \frac{dt}{\sqrt{1-t^4}}$$ is a new function. It is one of a family called the elliptic integrals, because one of them is the integral that defines the arc length of the ellipse. $$\int_0^x \frac{dt}{\sqrt{1-t^4}}$$ is the simplest elliptic integral to investigate, and many of its properties were found by analogy with those of the arcsine integral $$\int_0^x \frac{dt}{\sqrt{1-t^2}}$$. However, these were feats of virtuosity, like finding properties of the arcsine integral without using the sine function.The real innovation came around 1800, when Gauss realized that one should not study the elliptic integral $$u = \int_0^x \frac{dt}{\sqrt{1-t^4}}$$ but rather its inverse function x as a function of u (just as one should study the sine function rather than the arcsine integral). He wrote x = sl(u) and found that sl, like the sine function, is periodic:
$$sl(u + 2 \varpi) = sl(u), \quad {\rm where} \ \varpi \ \hbox{is a certain real number}$$
. More surprisingly, sl has second period 2iϖ, so sl is better viewed as a function of complex numbers. These results first became widely known when they were rediscovered and published by Abel and Jacobi in the 1820s. Further insights into double periodicity were obtained in the 1850s, as we will see in Chapter 16.

## Keywords

Elliptic Function Sine Function Polar Equation Elliptic Integral Addition Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Ayoub, R. (1984). The lemniscate and Fagnano’s contributions to elliptic integrals. Arch. Hist. Exact Sci. 29(2), 131–149.
2. Clebsch, A. (1864). Über einen Satz von Steiner und einige Punkte der Theorie der Curven dritter Ordnung. J. reine und angew. Math. 63, 94–121.
3. Legendre, A.-M. (1825). Traité des fonctions elliptiques. Paris: Huzard-Courcier.Google Scholar
4. Liouville, J. (1833). Mémoire sur les transcendantes elliptiques de première et de seconde espèce considérées comme fonctions de leur amplitude. J. Éc. Polytech. 23, 37–83.Google Scholar
5. Ore, O. (1957). Niels Henrik Abel: Mathematician Extraordinary. Minneapolis, MN.: University of Minnesota Press.
6. Rosen, M. (1981). Abel’s theorem on the lemniscate. Amer. Math. Monthly 88(6), 387–395.