Page 119 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Observe that the denominator of the integrand is never zero:
|f(z) + tg(z)|  ||f(t) – t|g(t)||  ||f(t)| – |g(t)|| > 0.

Observe that ¸ is continuous on the interval [0,1] and is integer-valued—t) is the
number of zeros of f + tg inside C. Being continuous and integer-valued on the connected
set [0,1], it must be constant. In particular, (0) = (1). This does the job!
1 f'(z)
 (t)   dz
2 i C f(z)

is the number of zeros of f inside C, and
1 f'(z) g'(t)

 (t)   dz.
2 i C f(z) g(z)


is the number of zeros of f + g inside C.
11.4 Keyword

Rouche’s Theorem: Suppose f and g are analytic on and inside a simple closed contour C. Suppose
moreover that |f(z)| > |g(z)| for all z  C. Then we shall see that f and f + g have the same
number of zeros inside C. This result is Rouche’s Theorem.

11.5 Self Assessment

1. Let C be a simple closed curve, and suppose f is analytic on C. Suppose moreover that the
only ................ of f inside C are poles.
2. The integer n is positive in case G is traversed in the ................, and negative in case the
traversal is in the negative direction.
3. Being continuous and integer-valued on the connected set [0,1], it must be constant. In
particular, ................

11.6 Review Questions

1. Let C be the unit circle |z| = 1 positively oriented, and let f be given by f(z) = z . How
3
many times does the curve f„°C„± wind around the origin? Explain.
2. Let C be the unit circle |z| = 1 positively oriented, and let f be given by

2
z  2
f(z)  .
z 3
How many times does the curve f(C) wind around the origin? Explain.
3. Let p(z) = a z + a z + ... + a z + a0, with a  0. Prove there is an R > 0 so that if C is the
n–1
n
n
n–1
1
n
circle |z| = R positively oriented, then
p'(z)
 p(z) dz  2n i.

C
4. How many solutions of 3e – z = 0 are in the disk |z| < 1? Explain.
z

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