Elementary Theory of Analytic Functions of One or Several Complex Variables by Henri Cartan

Author:Henri Cartan
Language: eng
Format: epub
Publisher: Dover Publications

4.ANALYTICITY OF HARMONIC FUNCTIONS

PROPOSITION 4.1.Any harmonic function g(x, y) in an open set D of the plane is an analytic function of the real variables x and y in D. In particular, any harmonic function is infinitely differentiable.

Proof.We can suppose that g has real values and, as the proposition is local (since it is sufficient to show that g is analytic in a neighbourhood of each point of D), we shall suppose that g(x, y) is harmonic in the open disc x2 + y2 < ρ2. In this disc, g is the real part of a holomorphic function f, which can be expanded as a power series

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