444 Chapter^7
whenever lz - z 01 < 8, and
f(z) = f(zo) - zof'(zo) + zf'(zo) + (z - zo)11(z) (7.10-7)
again for lz - zol < 8. The reader will note that (7.10-7) is also valid for
Z = Zo.
For n sufficiently large the rectangle Rn is contained in N 0 (z 0 ) (as soon
as dn < 8). Hence, by using (7.10-7) we obtain
J f(z) dz= [f(zo) - zof'(zo)] J dz+ f'(zo) J z dz
8Rn 8Rn 8Rn
+ [ (z - zo)11(z) dz
JaRn
= J (z - zo)11(z) dz
8Rn
since faRn dz= 0 and faRn zdz = 0 by Corollary 7.4 with f(z) = 1 and
f (z) = z. Therefore,
II(Rn)I = I J f(z) dzl = llaRn (z - zo)17(z) dz' :::; EdnLn
8Rn
where Ln is the length of the perimeter of Rn. But dn = d/2n and Ln =
L/2n, where L =length of the perimeter of R. Hence taking (7.10-2) into
account, we have
4-nlJ(R)I:::; IJ(Rn)I:::; 4-nEdL
or II(R)I :::; EdL, which implies that I(R) = 0 since E was arbitrary.
Lemma 7.2 If f is analytic in the disk lz - z 0 I < r, there exists a func-
tion F(z) analytic in the same disk such that F'(z) = J(z): i.e., f has a
primitive in a neighborhood of z 0 , called a local primitive off.
Proof For any point z in the disk let
F(z) = b1) t J(()d( = b2) t J(()d(
lzo lzo
(7.10-8)
where /l is the arc consisting of the horizontal side followed by the vertical
side of the rectangle with opposite vertices z 0 and z (Fig. 7.8), while 12 is
the arc consisting of the vertical side followed by the upper horizontal side
of the same rectangle. Clearly, the two integrals in (7.10-8) are equal by
the Cauchy-Goursat theorem for the rectangle (Lemma 7.1).
