1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

224 CHAPTER 6 • COMPLEX INTEGRATION

t Corollary 6.1 Let zo denote a fixed complex value. If C is a simple closed
contour with positive orientation such that z 0 lies interior to C, then

1

dz

2

.

--= 7Tt

e z-zo

r dz n = 0,

le (z - zo)

and

where n is any integer except n = 1.

Proof Since zo lies interior to C, we can choose R so that the circle CR with
center zo and radius R lies interior to C. Hence f (z) = (z-~o)" is analytic in a
domain D that contains both C and Cn and the region between them, as shown
in Figure 6.25.
We let CR have the parametrization

for 0 :s; 0 :s; 27T.

The deformation of contour theorem implies that the integral off over GR has
the same value as the integral off over C, so

1

--dz =^1 --dz = 12" --iRei9. d8 = i 12" dO = 27Ti

e z -zo e,. z - zo o Re^19 o

and

{ __ d_z_n = { dz n = {2" _iRe'o-dO = iR1- n {2" ei(l-n)9 dO

le (z -zo) le,. (z - zo) lo Rnein9 lo

= --Rl- n e•(l-n)9. 19=27f = _____ Rl- n Rl- n = 0.

1-n e=o 1-n 1-n

The deformation of contour theorem is an extension of the Cauchy- Goursat

theorem to a doubly connected domain in the following sense. We let D be

a domain that contains C 1 and C2 and the region between them, as shown in

y

x

Figure 6.25 The domain D that contains both C and CR.