The spherical harmonics
are the angular portion of the solution to Laplace's Equation in
Spherical Coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notational
convention being used. In the below equations, is taken as the azimuthal (longitudinal) coordinate, and as
the polar (latitudinal) coordinate (opposite the notation of Arfken 1985).

(1) |

(2) |

Integrals of the spherical harmonics are given by

(3) |

(4) | |||

(5) | |||

(6) |

where is a Legendre Polynomial.

The above illustrations show
(top) and
and
(bottom).
The first few spherical harmonics are

Written in terms of Cartesian Coordinates,

(7) | |||

(8) | |||

(9) |

so

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) | |||

(15) |

These can be separated into their Real and Imaginary Parts

(16) |

(17) |

The Zonal Harmonics are defined to be those of the form

(18) |

(19) |

(20) |

(21) |

(22) |

The spherical harmonics form a Complete Orthonormal Basis, so an arbitrary
Real function
can be expanded in terms of Complex spherical
harmonics

(23) |

(24) |

**References**

Arfken, G. ``Spherical Harmonics.'' §12.6 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 680-685, 1985.

Ferrers, N. M. *An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them.* London: Macmillan, 1877.

Groemer, H. *Geometric Applications of Fourier Series and Spherical Harmonics.*
New York: Cambridge University Press, 1996.

Hobson, E. W. *The Theory of Spherical and Ellipsoidal Harmonics.* New York: Chelsea, 1955.

MacRobert, T. M. and Sneddon, I. N. *Spherical Harmonics: An Elementary Treatise on Harmonic Functions,
with Applications, 3rd ed. rev.* Oxford, England: Pergamon Press, 1967.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Spherical Harmonics.'' §6.8 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 246-248, 1992.

Sansone, G. ``Harmonic Polynomials and Spherical Harmonics,'' ``Integral Properties of Spherical Harmonics and the Addition
Theorem for Legendre Polynomials,'' and ``Completeness of Spherical Harmonics with Respect to Square Integrable Functions.''
§3.18-3.20 in *Orthogonal Functions, rev. English ed.* New York: Dover, pp. 253-272, 1991.

Sternberg, W. and Smith, T. L. *The Theory of Potential and Spherical Harmonics, 2nd ed.* Toronto: University of Toronto Press, 1946.

© 1996-9

1999-05-26