Page 87 - Calculus with Complex Numbers

flz) = an (z - c)n + Jn+l (z - c)''-F1 + . . . ,

where an # 0. Therefore
f'lz) nanlz - c)''-1 + (n + 1)tz,z+1 (z - c)'' + . . .
flz) an (z - c)n + Jn+l (z - c)n+1 + . . .
1 nan + (rl + 1)tzn+1 (z - c) + . . .
z - c an + an +1 (z - c) + .-
has a simple pole at z = c with residue n by the cover up rule (see Section 4.8).
The result follows.

8.3 Rouché's theorem

The following theorem due to Rouché (1862) enables us to say something aboutthe
distribution of the zeros of a given function by comparing it with another function
whose zeros are known.

Theorem 3 (Rouché's theorem) lf f (z), g (z) are differentiable inside and on

the closed contour y, and if I f (z) I > Ig (z) Ifor all z eEy, then f (z), flz) + g (z)

have the same number of zeros inside y .
Proof lnformally, we can add any çsmaller' function g (z) to f (z) without
changing the number of zeros inside the contour.
By the argument principle it will be suflicient to prove that

/''(c) + g'lz) u /' ./''(c)
.
/' ' ) + ( ) ' v /'(c) dz'
'

c
'

-
z
g
c
(
/

J .
.
/
Observe that
f'lz) + g'(z) f'lz) d d
fl - = log(.f(z) + g(z)) - log flz)
z) + g(z) flz) dz dz