1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

8.7 • THE ARGUMENT PRI NCIPLE AND ROUCHE'S THEOREM 331

v

Figure 8.11 The points Wk on the contour f ( C) that winds a.round 0.

the values arg"•-• Wk- 1 and arg"• wk- 1 will be the same, so that log"•-• Wk- 1 =
Jog<>• Wk-1·

We can now show why f c ~(W dz counts the number of times that f ( C)

winds around the origin. We parametrize C: z (t), for as; ts; b, and choose the
appropriate branches of Jog"• w, giving

1

f' (z) dz= t t• f' (z(t)) z' (t)dt

c f (z) k = I lt•-• f (z (t))

n

= 2.::: (log,.. [/ (z (tk))] - log"• [f (z (tk- 1))])

k = l

n

= 2.::: (log"• Wk - log"• Wk-1) ,

k=l

which we rewrite as

(8-36}

When we use the fact that Po = Pn., the first summation in Equation (8-

36) vanishes. The summation of the quantities 6.1/JJ, expresses the accumulated