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Advanced Mathematics for Applications by Andrea Prosperetti

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14

Spherical Harmonics

14.1 Introduction

In the previous chapter we were led to a consideration of the associated Legendre equation by attempting to represent, in spherical coordinates, the angular dependence of a function u(r, θ, φ) defined over the entire range of the angular variables, 0 ≤ θπ, 0 ≤ φ ≤ 2π. We started by expressing u in a Fourier series in φ

image

(14.1.1)

and then we looked for suitable functions of θ useful to expand Sm. In this way, we were led to the associated Legendre functions of the first kind image (cos θ). We ...

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