A zonal harmonic is a spherical harmonic of the form P_l(cos θ), i.e., one which reduces to a Legendre polynomial. These harmonics are termed "zonal" since the curves on a unit sphere (with center at the origin) on which P_l(cos θ) vanishes are l parallels of latitude which divide the surface into zones. Resolving P_l(cos θ) into factors linear in cos^2 θ, multiplied by cos θ when l is odd, then replacing cos θ by z/r allows the zonal harmonic r^l P_l(cos θ) 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.

Legendre polynomial | sectorial harmonic | spherical harmonic | tesseral harmonic | zonal polynomial