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# Zonal Harmonic

A zonal harmonic is a spherical harmonic of the form , i.e., one which reduces to a Legendre polynomial (Whittaker and Watson 1990, p. 302). These harmonics are termed "zonal" since the curves on a unit sphere (with center at the origin) on which vanishes are parallels of latitude which divide the surface into zones (Whittaker and Watson 1990, p. 392).

Resolving into factors linear in , multiplied by when is odd, then replacing by allows the zonal harmonic to be expressed as a product of factors linear in , , and , with the product multiplied by when is odd (Whittaker and Watson 1990, p. 1990).

Legendre Polynomial, Sectorial Harmonic, Spherical Harmonic, Tesseral Harmonic, Zonal Polynomial

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## References

Byerly, W. E. "Zonal Harmonics." Ch. 5 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 144-194, 1959.Hashiguchi, H. and Niki, N. "Algebraic Algorithm for Calculating Coefficients of Zonal Polynomials of Order Three." J. Japan. Soc. Comput. Statist. 10, 41-46, 1997.Kowata, A. and Wada, R. "Zonal Polynomials on the Space of Positive Definite Symmetric Matrices." Hiroshima Math. J. 22, 433-443, 1992.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Zonal Harmonic

## Cite this as:

Weisstein, Eric W. "Zonal Harmonic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZonalHarmonic.html