Elliptic Integrals

An elliptic integral is an integral of the form \(\displaystyle I=\int _{a}^{b}F\left ( x, \sqrt {P(x)}\right ) dx, \) where P is a polynomial of degree 3 or 4 whose roots are all simple and F is a rational function of two variables. In this chapter, we will first explain where the adjective elliptic comes from (Sect. 5.1). Then we will study the special case of the so-called complete elliptic integrals of the first and second kinds, closely connected to Gauss hypergeometric function (Sects. 5.2 and 5.3)). We will show how to compute the numerical value of the complete integral of the first kind by Gauss arithmetic-geometric mean algorithm (Sect. 5.4). Finally, in Sect. 5.5, we define the Jacobian elliptic functions and give examples of computation of elliptic integrals.