Zonal Harmonic

A zonal harmonic is a spherical harmonic of the form P_l(costheta), 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 P_l(costheta) vanishes are l parallels of latitude which divide the surface into zones (Whittaker and Watson 1990, p. 392).

Resolving P_l(costheta) into factors linear in cos^2theta, multiplied by costheta when l is odd, then replacing costheta by z/r allows the zonal harmonic r^lP_l(costheta) to be expressed as a product of factors linear in x^2, y^2, and z^2, with the product multiplied by z when l is odd (Whittaker and Watson 1990, p. 1990).

See also

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

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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 3×3 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.

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Zonal Harmonic

Cite this as:

Weisstein, Eric W. "Zonal Harmonic." From MathWorld--A Wolfram Web Resource.

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