Complex Analysis - Final exam
Duration: 90 minutes
Documents and electronic devices are forbidden.
All the answers should be properly justified and explained.
The percentage of the grade awarded on each question is specified between parenthesis.
Each question will be graded independently: do not hesitate to skip some of them.
Exercise 1 : (20 %) Let r, s ∈R>0. Let fbe an analytic function defined on D(0, r) and
gbe an analytic function defined on D(0, s). Prove that f+gis analytic on D(0,min(r, s))
where min(r, s) is the minimum of rand s.
Exercise 2 : We want to study the limit of
X
n∈N
zn.
1. (10 %) Prove that the function
f(z) =
∞
X
n=0
zn
is well defined and infinitely differentiable on D(0,1/2).
2. (5 %) Prove that, for z∈D(0,1/2),
f(z) = 1
1−z.
3. (10 %) Is it true that for every z∈Csuch that z6= 1
∞
X
n=0
zn=1
1−z?
Justify your answer.
4. (15 %) Let a∈C\ {0}. Using f, express the function fadefined around 0 by
fa(z) = 1
a−z
as an analytic function of the form
∞
X
n=0
bnzn.
Give an r∈Rwith r > 0 such that this series converges on D(0, r) (rcan depend
on a).
