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Examples

Examples of harmonic functions of two variables are:

Examples of harmonic functions of three variables are given in the table below with r 2 = x 2 + y 2 + z 2: {\displaystyle r^{2}=x^{2}+y^{2}+z^{2}:}

Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). On regions without boundaries, adding the real or imaginary part of any entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution approaches 0 as r approaches infinity. In this case, uniqueness follows by Liouville's theorem.

The singular points of the harmonic functions above are expressed as " charges " and " charge densities " using the terminology of electrostatics, and so the corresponding harmonic function will be proportional to the electrostatic potential due to these charge distributions. Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will yield another harmonic function.

Finally, examples of harmonic functions of n variables are: