Harmonic Function - Connections With Complex Function Theory

Connections With Complex Function Theory

The real and imaginary part of any holomorphic function yield harmonic functions on R2 (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function u on an open subset Ω of R2 is locally the real part of a holomorphic function. This is immediately seen observing that, writing z = x + iy, the complex function g(z) := ux − i uy is holomorphic in Ω because it satisfies the Cauchy–Riemann equations. Therefore, g has locally a primitive f, and u is the real part of f up to a constant, as ux is the real part of .

Although the above correspondence with holomorphic functions only holds for functions of two real variables, still harmonic functions in n variables 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 one holds for them in analogy to the corresponding theorems in complex functions theory.

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