Real and Complex Analysis (Rudin)

388 REAL AND COMPLEX ANALYSIS
Since g is a conformal mapping of Q onto D(O; II I a I), (7) shows that
4
I g(z) I 2r). (11)
Let robe a large circle with center at 0; (11) gives (by Cauchy's theorem) that
A.2«() = 21 . r (z - ()g(z) dz = b - (.
1tl Jro
Substitute this value of A.i() into (11). Then (1) shows that the function
(12)
is bounded as z--+ 00. Hence q>
q>(z) = [ Q«(, z) - z ~ (}z - ()3 (13)
z E Q (') D, then I z - (I < 2r, so (2) and (13) give
has a removable singularity at 00. If
I q>(z) I < 8r 3 I Q«(, z)1 + 4r2 < I,OOOr2. (14)
By the maximum modulus theorem, (14) holds for all z E Q. This proves (3).
IIII
20.3 Lemma Suppose f E C~(R2), the space of all continuously differentiable
functions in the plane, with compact support. Put
a =!(~+ i~).
2 ax ay
Then the following" Cauchy formula" holds:
f(z) = -! fT (afX() de d"
n JR2 (- Z
(1)
(2)