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Advanced Mathematics for Applications by Andrea Prosperetti

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13

The Legendre Equation

13.1 Introduction

In spherical polar coordinates the Laplacian operator has the form

image

(13.1.1)

Our objective here is to simplify the action of this operator by expanding u on a suitable set of basis of functions. Since, in polar coordinates, the range of the coordinate ϕ is 0 ≤ ϕ < 2π (p. 171), it makes sense to start by expanding u in a Fourier series in ϕ:

image

(13.1.2)

Upon substituting this expansion into ∇2u and interchanging differentiation and summation,1 we have

(13.1.3)

A consideration ...

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