Page 493 - Electromagnetics

They are also complete and can be used to represent any piecewise smooth function f (r)
according to

f (r) = a λ ψ λ (r),
λ
which converges in mean when

(r) dV
V f (r)ψ λ m
= .
a λ m 2
ψ (r) dV
V λ m
These properties are shared by the two-dimensional eigenvalue problem involving an open
surface S with boundary contour .
Spherical harmonics. We now inspect solutions to the two-dimensional eigenvalue
problem

λ
2
∇ Y(θ, φ) + Y(θ, φ) = 0
a 2
over the surface of a sphere of radius a. Since the sphere has no boundary contour, we
demand that Y(θ, φ) be bounded in θ and periodic in φ. In the next section we shall
apply separation of variables and show that

$
2n + 1 (n − m)! m jmφ
Y nm (θ, φ) = P (cos θ)e
n
4π (n + m)!
m
where λ = n(n +1). Note that Q does not appear as it is not bounded at θ = 0,π. The
n
functions Y nm are called spherical harmonics (sometimes zonal or tesseral harmonics,
depending on the values of n and m). As expressed above they are in orthonormal
form, because the orthogonality relationships for the exponential and associated Legendre
functions yield
π π
∗
Y (θ, φ)Y nm (θ, φ) sin θ dθ dφ = δ n n δ m m . (A.96)

n m
−π 0
As solutions to the Sturm–Liouville problem, these functions form a complete set on the
surface of a sphere. Hence they can be used to represent any piecewise smooth function
f (θ, φ) as
∞ n

f (θ, φ) = a nm Y nm (θ, φ),
n=0 m=−n
where
π π
∗
a nm = f (θ, φ)Y (θ, φ) sin θ dθ dφ
nm
−π 0
m
by (A.96). The summation index m ranges from −n to n because P = 0 for m > n.For
n
negative index we can use
m
∗
Y n,−m (θ, φ) = (−1) Y (θ, φ).
nm
Some properties of the spherical harmonics are tabulated in Appendix E.3.

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