Legendre Polynomials and Related Functions

Abstract

4.1 Introduction The Legendre polynomials are closely associated with physical phenomena for which spherical geometry is important. In particular, these polynomials first arose in the problem of expressing the newtonian potential of a conservative force field in an infinite series involving the distance variables of two points and their included central angle (see Sec. 4.2). Other similar problems dealing with either gravitational potentials or electrostatic potentials also lead to Legendre polynomials, as do certain steady-state heat conduction problems in spherical solids, and so forth. There exist a whole class of polynomial sets which have many properties in common and for which the Legendre polynomials represent the simplest example. Each polynomial set satisfies several recurrence formulas, is involved in numerous integral relationships, and forms the basis for series expansions resembling Fourier trigonometric series, where the sines and cosines are replaced by members of the polynomial set. Because of all the similarities in these polynomial sets and because the Legendre polynomials are the simplest such set, our development of the properties associated with the Legendre polynomials will be more extensive than similar developments in Chap. 5, where we introduce other polynomial sets. In addition to the Legendre polynomials, we present a brief discussion of the Legendre functions of the second kind and associated Legendre functions. The Legendre functions of the second kind arise as a second solution set of Legendre's equation (independent of the Legendre polynomials), and the associated functions are related to derivatives of the Legendre polynomials.