Harmonic function
Connections with complex function theory

The real and imaginary part of any holomorphic function yield harmonic functions on R 2 {\displaystyle \mathbb {R} ^{2}} (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function u on an open subset Ω of R 2 {\displaystyle \mathbb {R} ^{2}} is locally the real part of a holomorphic function. This is immediately seen observing that, writing z = x + i y, {\displaystyle z=x+iy,} the complex function g (z):= u x − i u y {\displaystyle g(z):=u_{x}-iu_{y}} is holomorphic in Ω because it satisfies the Cauchy–Riemann equations. Therefore, g locally has a primitive f, and u is the real part of f up to a constant, as u is the real part of f ′ = g. {\displaystyle f'=g.}

Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in 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.
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