(1) Any function which is a solution of Laplace's equation and which has continuous first and second derivatives. (2) A function satisfying Laplace's partial differential equation, i.e., a solution of that equation. Often referred to, when the meaning is clear, as a harmonic. Laplace's equation was first solved using Fourier series, the terms of which (sines and cosines) are harmonically related. The term harmonic function was subsequently applied to all solutions of the equation. Other functions less obviously harmonic than sines and cosines are also solutions. Among them are Bessel functions, Legendre functions, and Lame' functions. The kind of harmonic function most suitable as a solution depends on the coordinate system in which the equation is expressed. Fourier series are best for Cartesian coordinate systems, Bessel functions for cylindrical coordinate systems, Lamé functions for ellipsoidal coordinate systems and so on.