CHAPTER 12 - Elliptic Integrals and Functions

Abstract

This chapter describes the elliptic integrals and functions. An elliptic integral is an integral of the form ∫ R(x, √P(x))dx, in which R is a rational function of its arguments and P(x) is a third- or fourth-degree polynomial with distinct zeros. Every elliptic integral can be reduced to a sum of integrals expressible in terms of algebraic, trigonometric, inverse trigonometric, logarithmic, and exponential functions (the elementary functions), together with one or more of the three special types of integral. The integrals are said to be expressed in the Legendre normal form. The number k is called the modulus of the elliptic integral, and the number k' = √1- k2 is called the complementary modulus of the elliptic integral. It is usual to set k = sin α, to call a the modular angle, and m = k2= sin2 α the parameter. The number n is called the characteristic parameter of the elliptic integral of the third kind.