1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

8.7 • THE ARGUMENT PRINGIPLE AND R.oUCHE'S THEOREM 333

Remark 8.4 Letting f (z) = z in Theorem 8.9 gives

W(C,a)= -^1 1 1 { 1
2

. --dz=
7r?. c z - a^0

if a is inside C, or

if a is outside C,

which counts the number of times the curve C winds around the point a. If C is
not a simple closed curve, but crosses itself perhaps several times, we can show
(but omit the proof) that W ( C, a) still gives the number of times the curve C
winds around the point a. Thus, winding number is indeed an appropriate term.

We close this section with a result that will help us gain information about
the location of the zeros and poles of meromorphic functions.