LECTURES ON THE EDGE-OF-THE-WEDGE THEOREM
5
Proof. For ae Ù, define
Fiz) - * (l^Ldw.
Im Ja w-z
F is evidently holomorphic in Ù. If ae Ù
÷
then
2rri
ËÏ
1
and
(2) 0 » f M * .
À
J u; - ae
Adding (1) and (2) we get F (z) = f (z) in Ù
ñ
since
/ - / /
du anj
dü2
Similarly, F(z) = / ( z ) in Ù
2
· Q.E.D.
The Cauchy formula (1) is used here in a Situation which is perhaps not quite
Standard, since the contour is merely rectifiable and / is not assumed to be holomor-
phic on it. Rather than go through the messy process of approximating 9Ù, from in-
side Ù
1
by smooth curves, it is easier to observe that (1) is certainly true when / is
a polynomial, and then to use the fact that the given / can be uniformly approximated
by polynomials on the closure of Ù
÷
. The well-known theorem of Mergelyan [Àè] may
be quoted here; an earlier theorem of Walsh will also do. Á similar remark applies to
(2).
The theorem is not true for arbitrary arcs à (i.e., rectifiability of à cannot be
dropped from the hypotheses). Interesting examples of this may be found in Denjoy's
paper[7].
In 1869, Schwarz [12] used the Cauchy theorem in exactly the above manner to
prove his reflection theorem, under the hypothesis that / is continuous on Å U W ,
holomorphic in W , real on E. Defining / in W~ by f (º) = f(z), it is clear that this
reflection theorem is a special case of Theorem A.
In more modern Statements of the reflection theorem it is usually only assumed
that f=u + iv is holomorphic in W , that í is continuous on EU W , and that í = 0
on Å [l , p. 171], [10, p. 230]. This is no longer a special case of Theorem A, since
the continuity of í does not a priori imply that u is also continuous on E. Other ver-
sions of the reflection theorem (involving Cluster sets) have been studied by Caratheo-
dory [6] .
The distribution versions of these theorems (i.e., Theorems  and C for ç = 1) do
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