3.12 Elliptic integrals

Jacobi introduced the complete elliptic integral of the second kind by the integral

where $\kappa$ is called the modulus. Thus Prop. 3.11 says that the perimeter of the ellipse is $4aE(\kappa)$ where $\kappa=\sqrt{1-\frac{b^{2}}{a^{2}}}$. (The constant $\sqrt{1-\frac{b^{2}}{a^{2}}}$ is called the eccentricity of the ellipse.)

It is important to note that elliptic integrals cannot be calculated precisely; the best we can do is to approximate $E(\kappa)$. We will now consider a method for estimating any proper definite integral.