LECTURES ON THE EDGE-OF-THE-WEDGE THEOREM
• Proof. For ae € Ù, define
Fiz) - * (l^Ldw.
Im Ja w-z
F is evidently holomorphic in Ù. If ae € Ù
(2) 0 » f M * .
J u; - ae
Adding (1) and (2) we get F (z) = f (z) in Ù
/ - / • / •
Similarly, F(z) = / ( z ) in Ù
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-
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
The theorem is not true for arbitrary arcs Ã (i.e., rectifiability of Ã cannot be
dropped from the hypotheses). Interesting examples of this may be found in Denjoy's
In 1869, Schwarz  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  .
The distribution versions of these theorems (i.e., Theorems Â and C for ç = 1) do