1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 457

Hence

(j + j) f(z)dz ~ (i +£ + 1+[,+1 +_[,

• f + j, + f + j, + f + j) f( z) dz ~ 0

which gives

J f(z)dz+ J f(z)dz+ J f(z)dz=O

0 -~ -~

or

J f(z) dz= J f(z)dz + J f(z) dz

0 01 02

since

J f(z) dz+ J f(z) dz= J f(z)dz

0' 011 0

J f(z)dz + J f(z)dz = J f(z) dz= -J f(z) dz

q q -~ ~

for k = 1,2.

As a particular case, consider any simple closed contour C and a circle

C1 such that Ci C Int C. If f is analytic on both C and Ci, as well as on

the doubly connected region R bounded by C* and c; (Fig. 7.13), we have

J f(z) dz= J f(z) dz (7.12-2)

0 01

provided that both C and C 1 are described the same number of times in
the same direction. This property reduces the evaluation of the integral

of f along C to the usually simpler evaluation of the integral along C1.

Formula (7.12-2) will be used often in what follows.
If we denote by ct the contour C 1 as described once in the positive
direction (a notation that we shall also apply to any contour later), and if