1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 475

case in which D is starlike, we may proceed as in the proof of Theorem

7.14 to obtain this result.

Cauchy's formula defines two different analytic functions, namely,

0.c(z)f(z) if z E IntC*, and the constant 0 if z E ExtC*. Thi~ shows

that the same analytic expression may define different functions on differ-

ent domains. Hence the specification of the domain must be considered

as an essential part of the definition of a function. On the other hand, it

should be noted that any given function may be expressed analytically or

"represented" in a variety of ways (by integrals, series, infinite products,

etc.)

Corollary 7 .14 If the contour C in Theorem 7 .23 is described once in the

positive direction and z E Int C*, then 0.c(z) = 1 and we have

J(z) = ~ j J(()d( (7.17-3)

27fZ ( - Z

c+

This is the classical form of Cauchy's formula. It can also be expressed as

J

f(z)dz = 21fif(a) (7.17-4)

z-a
c+
by replacing ( by z and z by a. In this form the formula is often applied to
the evaluation of integrals whose integrands are of the form J(z)/(z - a),
or that can be brought into that form.

Examples

1. Let C: z = 2eit, 0 ::; t ::; 27f. Then

J

COS 1fZ d.

2

.

-- Z = 27fz COS 7f = - 7fZ

z-1

c

2. With C as in Example 1, evaluate

J

zdz

(z^2 + l)(z -3)
c
Consider the circles /1: z -i = re it, 0 ::; t ::; 27f, and 12 : z + i = re it, 0 ::;
t ~ 27f, with 0 < r < 1 (Fig. 7.21), and apply Theorem 7.15 to obtain

J

z dz J z dz J · z dz

(z2 + l)(z -3) = (z2 + l)(z -3) + (z^2 + l)(z - 3)

c 'Yl 'Y2

= J z/[(z + i)(: - 3)] .dz+ J z/[(z - i)(: -3)] dz

z-z. z+i

'Yl 'Y2