1550251515-Classical_Complex_Analysis__Gonzalez_

744 Chapter9

where, as before, N denotes the number of zeros and P the number of
poles off inside C (counting multiplicities). The notation Licargf(z) is
used to indicate the variation (or increment) of any particular value of the
argument of f(z) as z is made to describe the contour C (Fig. 9.32).

Proof From Theorem 9.16 we have

N - p = 1 J f' ( z) dz

21Ti f(z)

(9.15-1)

c+

Letting w = f(z), dw = f'(z) dz, and C' = f(C), formula (9.15-1) becomes

N-P=~fdw

21Ti w

(9.15-2)

C'

In fact, if C: z = z(t), a :5 t :5 /3, then C': w = f(z(t)), a :5 t :5 /3, so
that either integral reduces to

-^1 fp f'(z(t)) z'(t)dt

21Ti lo: f(z(t))

Now, by the definition of winding number, we have

N-P= -.^1 J --dw =f2c1(0)
27l"z w - 0
C'
Hence
Ac arg f(z) =fie, argw = 21Tf2cn(O) = 21T(N - P) (9.15-3)
More generally, if C is any closed contour (not necessarily simple),
homotopic to a point in R, described a number of times in either direc-

y v

C'

0 u

"' - p = 2

Fig. 9.32