The Theorem and Formula of Cauchy 201n-th order derivative /(") of / exists at every point in G and
as long as the closed disk {z : \z - (\ < p] lies entirely in G; recall that
0! = 1 and, for a positive integer k, k\ = 1 • 2 • ... • k. Using the result of
Exercise 4.2.17, we can replace the circle of radius p in (4.2.16) with any
simple, closed, piece-wise smooth curve, as long as the domain bounded by
that curve contains the point z and lies entirely in G.
An informal way to derive (4.2.16) from (4.2.11) is to differentiate
(4.2.11) n times, bring the derivative inside the integral, and observe that
dn(Q - z)~l/dzn = n!(C - z)~n~l. A rigorous argument could go by induc-
tion, with both the basis and induction step computations similar to the
proof of (4.2.11). This proof is rather lengthy and does not introduce any-
thing new to our discussion; we leave the details to the interested reader.
Later, we will discuss an alternative rigorous derivation of (4.2.16) using
power series; see Exercise 4.3.8(c) on page 211 below.
The consequences of (4.2.16) are far-reaching indeed, as demonstrated
by the following results. The best part is, you can now easily prove all of
them yourself.
EXERCISE 4.2.20. (a)B Complete the proof of the Cauchy Integral Theo-
rem by showing that a continuous function that has the path-independence
property in a simply connected domain is analytic there. Hint: by Theorem
4-2.2, there exists an analytic function F so that F'(z) = f(z) for all z in G. By
(4-2.16), all derivatives of F are continuous in G, and you have f'{z) = F"(z).
(bf Use (4.2.16) to prove Cauchy's Inequality: 7/1/(01 < M when
\Q — z\ = p, then
\fin\z)\ < ^1. (4.2.17)
pn
Hint: use (4-2.8). Also note that since the function f is continuous, such a
number M always exists. (c)B Now use (4-2.17) to prove Liouville's
Theorem: A bounded entire function is constant. (That is, if a function is
analytic everywhere in the complex plain and is bounded, then the function
must be constant.) Hint: Taking n = 1 and p arbitrarily large, you conclude
that f'(z) = 0 for all z. Then recall part (c) of Exercise 4-2.8- (d)B Finally,
use Liouville's Theorem to prove the Fundamental Theorem of Algebra:
every polynomial of degree n > 1 with complex coefficients has at least one