Mathematics of Physics and Engineering

The Theorem and Formula of Cauchy 197

dence property in a simply connected domain has an anti- derivative there.

Theorem 4.2.2 Assume that a continuous function f = f(z) has the
path independence property in a domain G, and assume that the domain G
is simply connected (see page 186). Then there exists an analytic function
F = F(z) such that F'(z) = f(z) in G.

Proof. Define the function F(z) = Jc,z z* /(()d£, where C(zo, z) is a curve

in G, starting at a fixed point ZQ £ G and ending at z £ G. By Exercise

4.2.15 this function is well defined, because the integral depends only on

the points ZQ,Z and not on the particular curve. Define the number A =

A(z, Az) by

A_Fi,+££-m_m (4. 2 .9)

We need to show that limA 2 -+o A = 0 for every z £ G, which is equivalent
to proving that F'(z) = f(z). The result will also imply that F is analytic
inG.
Let C = C(z, z + Az) be any curve that starts at z and ends at z + Az.
Using the result of Exercise 4.2.16, we have f(z) = f(z)(l/Az) fcd£ =
(1/Az) fc f(z)dC, because f(z) is constant on C. Therefore,

^z JC(z,z+Az)
and, by (4.2.8), \A\ < (Lc/\Az)max(£c |/(C) - f(z)- Since we are free to
choose the curve C(z, z + Az), let it be the line segment from the point z
to the point z + Az. Then Lc = \Az. Since / is continuous at z, for every
e > 0, there exists a 6 > 0 so that \Az\ < 6 implies |/(C) — f(z)\ < e for all
C, £ C. As a result, for those Az, \A\ < e, which completes the proof. •

We will now state the first main result of this section, the Integral Theo-
rem of Cauchy, which says that the functions having the path independence
property in a simply connected domain are exactly the analytic functions.

Theorem 4.2.3 (The Integral Theorem of Cauchy) A continuous
function f has the path independence property in a simply connected domain
G if and only if f is analytic in G.

In Exercise 4.2.14, you already proved this theorem in one direction
(analyticity implies path independence) under an additional assumption
that /' is continuous; this is exactly what Cauchy did in 1825. This is not