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 * .

2ç 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 Ã (i.e., rectifiability of Ã 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 Â and C for ç = 1) do