Editing Harmonic function (section)

== Properties ==

The set of harmonic functions on a given open set {{mvar|U}} can be seen as the [[Kernel (linear operator)|kernel]] of the [[Laplace operator]] {{math|Δ}} and is therefore a [[vector space]] over {{tmath|\mathbb R\! :}} linear combinations of harmonic functions are again harmonic.

If {{mvar|f}} is a harmonic function on {{mvar|U}}, then all [[partial derivative]]s of {{mvar|f}} are also harmonic functions on {{mvar|U}}. The Laplace operator {{math|Δ}} and the partial derivative operator will commute on this class of functions.

In several ways, the harmonic functions are real analogues to [[holomorphic function]]s. All harmonic functions are [[analytic function|analytic]], that is, they can be locally expressed as [[power series]]. This is a general fact about [[elliptic operator]]s, of which the Laplacian is a major example.

The uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because every continuous function satisfying the mean value property is harmonic. Consider the sequence on {{tmath|(-\infty,0) \times \mathbb R}} defined by <math display="inline">f_n(x,y) = \frac 1 n \exp(nx)\cos(ny);</math> this sequence is harmonic and converges uniformly to the zero function; however note that the partial derivatives are not uniformly convergent to the zero function (the derivative of the zero function). This  example shows the importance of relying on the mean value property and continuity to argue that the limit is harmonic.