Elliptic functions

An elliptic integral is an integral of the form

$$\int R(\sqrt{P(x)}, x) \; dx$$

where $P(x)$ is a cubic or quartic polynomial in $x$ and $R(x,y)$ is a rational function of $x$ and $y$. Except for some trivial cases, elliptic integrals cannot be reduced to expressions that only involve elementary functions. Legendre did succeed in showing that every elliptic integral can be reduced to an expression involving elementary functions and three particular kinds of elliptic integral. Of these three kinds, the simplest can be written in the following form:

$$F(x\vert k) = \int_0^x \frac{dx}{\sqrt{(1-x^2)(1-k^2 x^2)}}$$

where $k$ is viewed as a parameter called the modulus.

Around 1825 to 1827, Carl Jacobi and Niels Abel had an idea that would revolutionize the treatment of problems involving what are now known as elliptic integrals. The idea centered on viewing the expression $F(x\vert k)$ as analogous to an inverse sine function. Incidentally, as it turns out, Gauss came up with the idea of inverting elliptic integrals about 20 years before Jacobi and Abel - but Gauss did not publish any of this work. Gauss did however publish some work that depended on his elliptic functions work - enough that Jacobi knew that he and Abel were not the first to tread on this ground.

In calculus, students are taught that

$$\sin^{-1} x = \int_0^x \frac{dx}{\sqrt{1-x^2}}$$

provided that $-1<x<1$. Imagine having to develop trigonometry starting with this integral. Now imagine that a bright student gets the idea of defining

$$x = \sin u$$

and working not with the integral but instead with $\sin u$.

Essentially the idea that Abel and Jacobi found fruitful [Jacobi1829, §17] was to say that if

$$u = F(x\vert k)$$

then instead work with

$$x = {\rm sn}(u,k)$$

In the trigonometric case, it helps to define the cosine function. In the elliptic functions game, it helps to have two separate analogues of the cosine function, corresponding to the quadratic factors in the elliptic integral:

$${\rm cn}^2(u,k) = 1-{\rm sn}^2(u,k)$$

and

$${\rm dn}^2(u,k) = 1-k^2{\rm sn}^2(u,k)$$

The elliptic functions sn, cn and dn obey a system of differential equations analogous to the trig functions:

$$\frac{\partial}{\partial u} {\rm sn}(u,k) = {\rm cn}(u,k) \; {\rm dn}(u,k)$$

$$\frac{\partial}{\partial u} {\rm cn}(u,k) = - {\rm sn}(u,k) \; {\rm dn}(u,k)$$

$$\frac{\partial}{\partial u} {\rm dn}(u,k) = -k^2 {\rm sn}(u,k) \; {\rm cn}(u,k)$$

with initial conditions

$${\rm sn}(0,k) = 0$$

$${\rm cn}(0,k) = {\rm dn}(0,k) = 1$$

The elliptic functions provide generalizations of the usual trigonometric identities, such as the angle addition formulas, but we omit those developments here.