Real and Complex Analysis (Rudin)

236 REAL AND COMPLEX ANALYSIS
We may summarize this by saying that every real harmonic function is locally
the real part of a holomorphic function.
Consequently, every harmonic function has continuous partial derivatives of
all orders.
The Poisson integral also yields information about sequences of harmonic
functions:
11.11 Harnack's Theorem Let {u.} be a sequence of harmonic functions in a
region n.
(a) If u. -4 u uniformly on compact subsets ofn, then u is harmonic in n.
(b) If U 1 ~ U2 ~ U3 ~ ... , then eit"her {u.} converges uniformly on compact
subsets ofn, or u.(z)-4 00 for every ZEn.
PROOF To prove (a), assume D(a; R) c n, and replace u by u. in the Poisson
integral 11.10(1). Since u. -4 u uniformly on the boundary of D(a; R), we conclude
that u itself satisfies 11.10(1) in D(a; R).
In the proof of (b), we may assume that U 1 :2= O. (If not, replace u. by
u. - u 1 .) Put u = sup u., let A = {z E n: u(z) < oo}, and B = n - A. Choose
D(a; R) en. The Poisson kernel satisfies the inequalities
R - r R2 - r2 R + r
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