Editing Harmonic function (section)

== Connections with complex function theory ==
The real and imaginary part of any holomorphic function yield harmonic functions on {{tmath|\mathbb R^2}} (these are said to be a pair of [[harmonic conjugate]] functions). Conversely, any harmonic function {{mvar|u}} on an open subset {{math|Ω}} of {{tmath|\mathbb R^2}}  is ''locally'' the real part of a holomorphic function. This is immediately seen observing that, writing <math>z = x + iy,</math> the complex function <math>g(z) := u_x - i u_y</math> is holomorphic in {{math|Ω}} because it satisfies the [[Cauchy–Riemann equations]]. Therefore, {{mvar|g}}  locally has a primitive {{mvar|f}}, and {{mvar|u}} is the real part of {{mvar|f}} up to a constant,  as {{mvar|u{{sub|x}}}} is the real part of <math>f' = g.</math>

Although the above correspondence with holomorphic functions only holds for functions of  two real variables, harmonic functions in {{mvar|n}} variables still enjoy a number of properties typical of holomorphic functions. They are (real) analytic; they have a maximum principle and a mean-value principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions theory.