Editing Harmonic function (section)

== Examples ==
Examples of harmonic functions of two variables are:
* The real or imaginary part of any [[holomorphic function]].
* The function <math>\,\! f(x, y) = e^{x} \sin y;</math> this is a special case of the example above, as <math>f(x, y) = \operatorname{Im}\left(e^{x+iy}\right) ,</math> and <math>e^{x+iy}</math> is a [[holomorphic function]]. The second derivative with respect to ''x'' is <math>\,\! e^{x} \sin y,</math> while the second derivative with respect to ''y'' is <math>\,\! -e^{x} \sin y.</math>
* The function <math>\,\! f(x, y) = \ln \left(x^2 + y^2\right)</math> defined on <math>\mathbb{R}^2 \smallsetminus \lbrace 0 \rbrace .</math> This can describe the electric potential due to a line charge or the gravity potential due to a long cylindrical mass.

Examples of harmonic functions of three variables are given in the table below with <math>r^2=x^2+y^2+z^2:</math>

:{| class="wikitable"
! Function !! [[Mathematical singularity|Singularity]]
|-
|align=center|<math>\frac{1}{r}</math>
|Unit point charge at origin
|-
|align=center|<math>\frac{x}{r^3}</math>
|''x''-directed dipole at origin
|-
|align=center|<math>-\ln\left(r^2 - z^2\right)\,</math>
|Line of unit charge density on entire z-axis
|-
|align=center|<math>-\ln(r + z)\,</math>
|Line of unit charge density on negative z-axis
|-
|align=center|<math>\frac{x}{r^2 - z^2}\,</math>
|Line of ''x''-directed dipoles on entire ''z'' axis
|-
|align=center|<math>\frac{x}{r(r + z)}\,</math>
|Line of ''x''-directed dipoles on negative ''z'' axis
|}

Harmonic functions that arise in physics are determined by their [[mathematical singularity|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 (complex analysis)|Liouville's theorem]].

The singular points of the harmonic functions above are expressed as "[[Charge (physics)|charges]]" and "[[Charge density|charge densities]]" using the terminology of [[electrostatics]], and so the corresponding harmonic function will be proportional to the [[Electric potential|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 [[Method of inversion|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 {{mvar|n}} variables are:

* The constant, linear and affine functions on all of {{tmath|\mathbb R^n}} (for example, the [[electric potential]] between the plates of a [[capacitor]], and the [[gravity potential]] of a slab)
* The function <math> f(x_1, \dots, x_n) = \left({x_1}^2 + \cdots + {x_n}^2\right)^{1-n/2}</math> on <math>\mathbb{R}^n \smallsetminus \lbrace 0 \rbrace</math> for {{math|''n'' > 2}}.