Spherical Harmonics

John Aieilo

Laplace's Equation, V2U=0, can be written in the spherical coordinate system of Section 2.3 as:

d2U 2 W I d2U , cotg 3t/ l 9*u dr2 r dr M2 r2 3» Vsin^ ^ ( '

If a trial substitution of U(r,9,<p) = R(r)Y(9,<l>) is made, the following equations are obtained through a separation of variables:

where n(n + 1) has been chosen as the separation constant. Solutions to Eq. (G-2) are of the form

Thus, solutions to Laplace's Equation (Eq. (G-l)) are of the form

U=[Arn + Br-<"+})]Y(9,4>), n=0,l,2,..., (G-5)

These functions are referred to as solid spherical harmonics, and the Y(0,<f>) are known as surface spherical harmonics. We wish to define U over a domain both interior and exterior to a spherical surface of radius r, and to have U continuous everywhere in the domain and to assume prescribed values 11^0,$) on the surface. Under these conditions, Eq. (G-5) with B=0 gives the form of U for the interior region of the sphere and with A = 0 represents its form in the exterior region. To determine the surface spherical harmonics, the trial substitution y(0,<f>)=P(cos0)<I>(<f>) (G-6)

Multiplying by sinty/P<I> and choosing a separation constant of m2 yields d2P(cos9) dPicosO)

The solutions to Eq. (G-8) are readily found to be spherical harmonics «!>(<>)= C cos m<f>+ S sin m<t>

in which m must be an integer, because 4>(<f>) is required to be a single valued function. Equation (G-7) can be rewritten substituting x = cos 9 as.

l-x2

which is the generalized Legendre equation [Jackson, 1962]. For m = 0, the solutions to Eq. (G-10) are called Legendre polynomials and may be computed from either Rodrigues' formula

or from a recurrence relation convenient for computer use [Arfken, 1970],

.(*) = 2xPn(X) - P„_ ,(x)- [xP„(x)-1»._,(*)]/(« + 1) (G-12)

Rodrigues' formula can be verified by direct substitution into Eq. (G-10), and the recurrence relation can be verified by mathematical induction. When m ± 0, solutions to Eq. (G-10) are known as associated Legendre functions (of degree, n, and order, m), and may be computed by [Yevtushenko, et al., 1969]

or by [Heiskanen and Moritz, 1967]

dx"

where / is either (n — m)/2 or (n — m — l)/2, whichever is an integer. Table G-l lists the associated Legendre functions up to degree and order 4 in terms of cos 9 [Fitzpatrick. 1970].»

Table G-I. Explicit Forms of Associated Legendre Functions Through Degree n = 4 and Order m=>4
0 0

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