Elliptic Functions

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: . 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.