Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

#### *(c) Copyright 1994,1995,1996 by Evans M. Harrell
II and James V. Herod. All rights reserved.

version of 30 August 1996

Further discussion of the
Legendre polynomials and spherical harmonics

While Maple or
Mathematica
can take care of doing calculations with special functions
like the Legendre polynomials, it is helpful to remember where they came from,
in order to be able to set calculations up properly. In problems like the one
we began with, these special functions occur in the combination

where the N's are normalization factors which we may choose at our convenience.
Because these come from a self-adjoint eigenvalue problem, they form an
orthogonal set, with respect to integration over the angular variables, so the
N's can be chosen to make them orthonormal. The result is called a
*spherical harmonic * and denoted

(11.8)

it is known to Mathematica, of course, as
`SphericalHarmonicY[l,m,theta,phi]` .

The spherical harmonics are eigenfunctions of the angular part of the Laplace
operator, known to physicists as the angular momentum operator:

(11.9)

Notice that the eigenvalue does not depend on m, but m does show up in one way,
because the ranges of the indices are

*l* = 0, 1, 2, ...; -*l* <= m <= *l* .

For each eigenvalue there are 2*l* + 1 different eigenfunctions,
corresponding to the number of values of m between -*l* and *l* .

Because the solid angle element on the sphere is sin(theta) d theta d phi,
the orthonormality relationship is:

(11.10)

which is 0 unless both m=m' and *l * = *l' *. Don't overlook the
complex conjugation of the second spherical harmonic, which amounts to changing
its exp(imphi) to exp(-i m phi). If we rewrite
(11.9) in terms of the
associated
Legendre polynomials and the variable z - notice that
sin(theta) d theta = - dz - we find

which incidentally shows what the orthogonality relationship is for the
associated (and usual) Legendre polynomials:

(11.11)

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