Mathematics of Physics and Engineering

The Theorem and Formula of Cauchy 199

Proof. Remember that the change of the orientation reverses the sign of
the line integral; see Exercise 4.2.12, page 195. Connect the curves Cfc,
k = 1,... ,n to Co with smooth curves (for example, line segments), and
apply the Integral Theorem of Cauchy to the resulting simply connected
domain (draw a picture). Then note that the integrals over the connecting
curves vanish (you get two for each, with opposite signs due to opposite
orientations), and you are left with (4.2.10). •

EXERCISE 4.2.17? Use (4.2.10) to show that if f is analytic in G andCi,C 2
are two simple closed curves in G with the same orientation (both clockwise
or both counterclockwise) so that Ci is in the domain enclosed by C\, then
§c f(z)dz = §c f(z)dz. Note that G does not have to be simply connected.
Hint: consider the domain G\ that lies in between C\ and C2.

We now use the "analyticity implies path independence" part of the In-
tegral Theorem of Cauchy to establish the Integral Formula of Cauchy.

Theorem 4.2.5 (Integral Formula of Cauchy) Let f = f(z) be an

analytic function in a simply connected domain G, and C, a simple closed

curve in G, oriented counterclockwise. Then the equality

i^-hfM* <

4

'

211

»

holds for every point z inside the domain bounded by C.

Proof. Step 1. Fix the curve C and the point z. Since / is analytic in
G, the function /(£) = /(C)/(C — z) is analytic in the domain G with the
point z removed. Let Cp be a circle with center at z and radius p small
enough so that Cp lies completely inside the domain bounded by C. We
orient the circle counterclockwise and use the result of Exercise 4.2.17 to
conclude that §c f{QdC, = §Cp f(()d(.

Step 2. By (4.2.6), f{z) = (1/2TTJ) JC (f(z)/{C, - z))d( (keep in mind
that the variable of integration is (, whereas z is fixed). Combining this
with the result of Step 1, we find:

^iP^dC- m = -L / /«>-/(*>«. (4.2.12)

2-Ki Jc C - z S M ; 2ni JCp C-z V ;

Step 3. Note that the left-hand side of (4.2.12) does not depend on p.