Real and Complex Analysis (Rudin)

238 REAL AND COMPLEX ANALYSIS
Suppose J = u + iv is holomorphic in n+, and
(2)
Jor every sequence {z.} in n+ which converges to a point oj L.
Then there is a Junction F, holomorphic in n+ u L u n-, such that
F(z) = J(z) in n+; this F satisfies the relation
F(z) = F(z) (z E n+ u L u n-). (3)
The theorem asserts that J can be extended to a function which is holomorphic
in a region symmetric with respect to the real axis, and (3) states that F
preserves this symmetry. Note that the continuity hypothesis (2) is merely
imposed on the imaginary part off
PROOF Put n = n+ u L u n-. We extend v to n by defining v(z) = 0 for
z ELand v(z) = - v(z) for ZEn -. It is then immediate that v is continuous
and that v has the mean value property in n, no that v is harmonic in n, by
Theorem 11.13.
Hence v is locally the imaginary part of a holomorphic function. This
means that to each of the discs D t there corresponds an J; E H(D t ) such that
1m J; = v. Each J; is determined by v up to a real additive constant. If this
constant is chosen so that J;(z) = J(z) for some z E D t (') II +, the same will
hold for all z E D t (') II+, sinceJ - J; is constant in the region D t (') II+. We
assume that the functions J; are so adjusted.
The power series expansion of J; in powers of z - t has only real coefficients,
since v = 0 on L, so that all derivatives of J; are real at t. It follows
that
J;(z) = J;(z) (z EDt). (4)
Next, assume that D. (') D t "# 0. ThenJ; = J = J. in D t n D. (') II+; and
since D t n D. is connected, Theorem 10.18 shows that
J;(z) = J.(z)
(z E D t (') D.).
(5)
Thus it is consistent to define
F(z) =
J(z)
J;(z)
J(z)
1
for z E n+
for z E D t
for z E n-
and it remains to show that F is hoi om orphic in n-. If D(a; r) c: n-, then
D(a; r) c: n+, so for every z E D(a; r) we have
co
J(z) = L c.(z - a)·. (7)
.=0
(6)