ABSTRACT
This section and the next are entirely devoted to the technical task of defining the Legendre polynomials and establishing a number of their special properties. It is natural to wonder about the purpose of this elaborate machinery, and more generally, why we care about Legendre polynomials at all. The simplest answer is that the Legendre polynomials have many important applications to mathematical physics, and these applications depend on this machinery. For the benefit of readers who wish to see for themselves, the physical background and several typical applications are discussed in Appendix A. There is another answer, however, which is less utilitarian and applies equally to our subsequent treatment of Bessel functions. It is that the study of specific classical functions and their individual properties provides a healthy counterpoise to the abstract ideas that sometimes seem to dominate contemporary mathematics. In addition, we mention several items that arise naturally in the context of this chapter which we hope will be of interest to all students of mathematics: the gamma function and the formula tan x = 1 1 x − 1 3 x − 1 5 x − ⋯ ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315371825/895247ed-bf4c-467d-b370-ced59759b8c3/content/eq1704.tif"/>
