Page 85 - Calculus with Complex Numbers

8 . 2 A rg u m e n t p r i n c i p I e
W e can count the number of zeros a function has inside a closed contour by means
of the following theorem.

Theorem 2 (Argument principle) lf flz) is differentiable inside and on the
closed contour y, and if f (z) # 0 anywhere on y, then the number N of zeros of
f (z) inside y is given by the formula
N 1 . /''(c)
= Jz.
ln'i y flz)
Geometrical interpretation Observe that

v
1 f'lz) 1
'
lxi l . /'(z) '''' ''-' l n'i 7''VEZCZII'Z

D
since log flz) is a primitive of f' lzl(f (z). But (see Section 2. 12)

and log I flz) I is single valued. Therefore,
1 s a 1
ln'i Llog .f(z)1), = z zv Larg .f(z)1), .

So Theorem 2 says that the number of zeros of flz) inside y is equal to the number
of times flz) circulates the origin as z goes round y.
:
Example Suppose flz) = .2 - 1.

yl = circle centre 0 radius 1/2.

We can parametrise n as z = eit/l (0 :jq t :jq 2:7r). Therefore, the image
contotlr /'(y1 ) parametrises as w = f (z) = .2 - 1 = elitjzî - 1 which is the
:
.
circle centre - 1 radius 1/4 described twice. Observe that ./'(y1 ) does not circulate
the origin at all corresponding to the fact that there are no zeros of f (z) inside n
(Figure 8.1).

Ffgure 8. /