Real and Complex Analysis (Rudin)

ELEMENTARY PROPERTIES OF HOLOMORPHIC FUNCTIONS 219
then
1 i f(w)
f(z) . Indr (z) = -. -- dw
2m r w - z
for ZEn - r*
(2)
and
Lf(Z) dz = o.
If r 0 and r 1 are cycles in n such that
Indro (IX) = Indrl (IX) for every IX not in n,
(3)
(4)
then
r f(z) dz = r f(z) dz.
Jro Jrl
(5)
PROOF The function g defined in n x n by
If(W) - f(z)
if w "# z,
g(z, w) = w - z
f'(z) if w = z,
is continuous in n x n (Lemma 10.29). Hence we can define
(6)
h(z) = -2 1 . r g(z, w) dw
mJr
(z En).
For ZEn - r*, the Cauchy formula (2) is clearly equivalent to the assertion
that
(7)
h(z) = O. (8)
To prove (8), let us first prove that hE H(n). Note that g is uniformly
continuous on every compact subset of n x n. If ZEn, Zn E n, and Zn- Z, it
follows that g(zn' w) - g(z, w) uniformly for w E r* (a compact subset of n).
Hence h(zn)- h(z). This proves that h is continuous in n. Let A be a closed
triangle in n. Then
1
h(z) dz = 21. r (1 g(z, w) dZ) dw.
M m Jr a&
For each WEn, z- g(z, w) is holomorphic in n. (The singularity at z = w is
removable.) The inner integral on the right side of (9) is .therefore 0 for every
WE r*. Morera's theorem shows now that hE H(n).
(9)