Harmonic Function - Examples

Examples

Examples of harmonic functions of two variables are:

  • The real and imaginary part of any holomorphic function
  • The function, this is a special case of the example above, as, and is a holomorphic function.
  • The function
defined on (e.g. the electric potential due to a line charge, and the gravity potential due to a long cylindrical mass)


Examples of harmonic functions of three variables are given in the table below with :

Function Singularity
Unit point charge at origin
x-directed dipole at origin
Line of unit charge density on entire z-axis
Line of unit charge density on negative z-axis
Line of x-directed dipoles on entire z axis
Line of x-directed dipoles on negative z axis
Harmonic functions are in part determined by their singularities. Adding the real or imaginary part of any entire function will produce a harmonic function with the same singularity; however, only one can go to 0 in modulus away from the boundary 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:

  • The constant, linear and affine functions on all of Rn (for example, the electric potential between the plates of a capacitor, and the gravity potential of a slab)
  • The function on for n > 2.

Read more about this topic:  Harmonic Function

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