Spherical harmonics
The spherical harmonics, denoted
, where is a nonnegative integer, and an integer lying in the range
, are the wellbehaved solutions to

(A.159) 
on the surface of a sphere (i.e., constant). Here,
denotes a Laplacian (Riley 1974a), and
, , are standard spherical coordinates. The spherical harmonics take the form (Jackson 1975)

(A.160) 
where the
are associated Legendre polynomials (Abramowitz and Stegun 1965a).
In particular,

(A.161) 
where the are the Legendre polynomials introduced in Section 3.4. The spherical harmonics satisfy

(A.162) 
and have the property that they are orthonormal when integrated over the surface of a sphere; that is,

(A.163) 
The first few spherical harmonics are:


(A.164) 


(A.165) 


(A.166) 


(A.167) 


(A.168) 


(A.169) 


(A.170) 


(A.171) 


(A.172) 