1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

320 CHAPTER 8 • RESIDUE THEORY

where g is analytic at z = to. Using the parametrization of Gr and Equation

(8-23), we get

1 1

" ir·e•o d8 1"..

f (z) dz= Res [f , to) --. 8 - + ir g (to+ re'^8 ) e'^8 d8

C r o re o

= i1rRes [f, to] + ir lo" g (to+ re•^8 ) e'^8 d8. (8-^24 )

A.$ g is continuous at t 0 , there is an M > 0 so that 19 (to+ rei^8 ) I :::; M, and

l

lim ir ("' g (to+ re•^8 ) ei^8 dOI :::; lim r {" MdO = lim r1rM = 0.

r-o lo r-0 lo r-o

Combining this inequality with Equation (8-24) gives the conclusion we want.

y

Figure 8.6 The poles ti, h,... , t1 off that lie on the x-axis and the poles z1, z2, ... ,

Zk that lie above the semicircles C1, C2, ... , C,.