Real and Complex Analysis (Rudin)

ELEMENTARY PROPERTIES OF HOLOMORPlDC FUNCTIONS 20S
PROOF If [ex, PJ is the parameter interval of y, the fundamental theorem of
calculus shows that
r
iF'(Z) dz = F'(y(t»y'(t) dt = F(y(P» - F(y(ex» = 0,
since y(P) = y(ex).
Corollary Since zn is the derivative of zn+ 1/(n + 1) for all integers n =Ihave
IIII
-1, we
for every closed path y if n = 0, 1, 2, ... , and for those closed paths y for which
o ¢ y* ifn = -2, -3, -4, ....
The case n = -1 was dealt with in Theorem 10.10.
10.13 Cauchy's Theorem for a Triangle Suppose .1 is a closed triangle in a
plane open set n, p E n,Jis continuous on n, andfE H(n - {p}). Then
r f(z) dz = O.
JM.
(1)
For the definition of iJ.1 we refer to Sec. 1O.9(c). We shall see later that our
hypothesis actually implies that f E H(n), i.e., that the exceptional point p is not
really exceptional. However, the above formulation of the theorem will be useful
in the proof of the Cauchy formula .
. PROOF We assume first that p ¢ .1. Let a, b, and c be the vertices of .1, let a',
b', and c' be the midpoints of [b, c], [c, a], and [a, b], respectively, and consider
the four triangles .1i formed by the ordered triples
{a, c', b'}, {b, a', c'}, {c, b', a'}, {a', b', c'}.
(2)
If J is the value of the integral (1), it follows from 10.9(6) that
J = it
id/(Z) dz.
(3)
The absolute value of at least one of the integrals on the right of (3) is therefore
at least I J141. Call the corresponding triangle .11, repeat the argument
with .11 in place of .1, and so forth. This generates a sequence of triangles .1n