Next: Perihelion precession of Mercury
Up: Useful mathematics
Previous: Matrix eigenvalue theory
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) 
and 



(A.172) 
Next: Perihelion precession of Mercury
Up: Useful mathematics
Previous: Matrix eigenvalue theory
Richard Fitzpatrick
20160331