In this chapter we consider one last variation, Legendre polynomials, on our theme of using complete orthogonal sets of functions to solve boundary value problems. These functions arise when we separate variables in spherical coordinates. We will only consider problems with all data independent of the polar angle θ.
The chapter begins with a general introduction to the idea of orthogonal polynomials. The most important point in this introduction is Lemma 6.1. The rest of the introduction is optional; read it if you find the very general picture it presents to be useful.
Section 6.2 defines the Legendre polynomials by a seemingly arbitrary formula, and finds many of their properties. If you prefer to start with the PDE problem, read p. 174 through one sentence after (6.17) on p. 175, then the first sentence on p. 169, then go back to the start of §6.2. Note: you found the Legendre Polynomials Pn(x) for n = 0, 1, & 2 in Exercise 7 of §3.2 from just the fact that they are orthogonal polynomials of degree n.
In §6.2, the most important results are Theorems 6.1, 6.4, and 6.6, and Corollary 6.1. Be sure you understand why the norms are different in 6.1 and 6.4. We will also use Theorems 6.2 and 6.3 (but not 6.5, only its corollary). In §6.3, you do not need to read beyond the first sentence after (6.17) on p. 175.
Optional: For a brief view of adding θ-dependence, continue on through p. 176, then skip to "A number of other boundary value problems ..." near the bottom of p. 181.
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Most recently updated on May 25, 2018.