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Special Functions by Roderick Wong, Richard Beals

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9

Spherical functions

Spherical functions are solutions of the equation

image

(9.0.1)

that arises from separating variables in Laplace’s equation Δu = 0 in spherical coordinates. Surface harmonics are the restrictions to the unit sphere of harmonic functions (solutions of Laplace’s equation) in three variables. For surface harmonics, m and ν are non-negative integers. The case m = 0 is Legendre’s equation

image

(9.0.2)

with ν a non-negative integer. The solutions to equation (9.0.1) that satisfy the associated boundary conditions are Legendre ...

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