Test 1
Due Oct. 1 at the beginning of class
This exam must be worked on independently. However, you are allowed to
use your notes, the course textbook, and old homework. You are also allowed to
talk to the instructor. You are
not
allowed to use any other books, or to talk to
anyone else about the exam.
Keep in mind that it is very easy to tell when a student has plagiarized a proof. Also keep in mind
that
the
University
of
Iowa
takes
academic
misconduct
very
seriously,
and
that
anyone
caught
cheating will receive, at minimum, a failing grade in this course, and will also be reported to the
dean for possible further disciplinary action.
1. (10 points) Suppose that
f
:
C
→
C
is entire and
f
(
z
) = 1 for all
z
∈
C
.
Prove that if
f
 
f

2
+ 3
f
3
= 7
,
then
f
is a constant function.
2. (15 points) Let
D
be a domain in
C
which is symmetric about the real
axis (i.e. if
x
+
iy
∈
D
, then
x

iy
∈
D
). Prove that if
f
is analytic and
nonconstant on
D
, then the function
f
(
z
) is not analytic on
D
.
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 Fall '09
 Forde
 let Cρ denote

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