A First Course in Complex Analysis · Problem 35

Isolated Singularities and the Residue Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · zeros-of-holomorphic-functions

Problem

Suppose

f

is holomorphic and not identically zero on an open disk

D

centered at

a

, and suppose

f(a) = 0

. Use the following outline to show that

\operatorname{Re} f(z) > 0

for some

z

D

. (a) Why can you write

f(z) = (z-a)^m g(z)

where

m > 0

g

is holomorphic, and

g(a) \neq 0

? (b) Write

g(a)

in polar coordinates as

c\, e^{i\alpha}

and define

G(z) = e^{-i\alpha} g(z)

. Why is

\operatorname{Re} G(a) > 0

? (c) Why is there a positive constant

\delta

so that

\operatorname{Re} G(z) > 0

for all

z \in D[a,\delta]

? (d) Write

z = a + r e^{i\theta}

for

0 < r < \delta

. Show that

f(z) = r^m e^{im\theta} e^{i\alpha} G(z)

. (e) Find a value of

\theta

so that

f(z)

has positive real part.