12 - Elliptic Integrals and Functions

Abstract

This chapter focuses on 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: (1) elliptic integral of the first kind, (2) elliptic integral of the second kind, and (3) elliptic integral of the third kind. These integrals are expressed in the Legendre normal form. The number k is called the modulus of the elliptic integral, and the number k' = √l —k2 is called the complementary modulus of the elliptic integral. The chapter also discusses the tabulations and trigonometric series representations of complete elliptic integrals.