Math 419 Exam 1, Fall 2010
This test is worth 100 points. Please indicate your solutions to the following problems as clearly as possible. All complex numbers should be expressed
in rectangular or polar form. Please be thorough – I will be awarding partial
credit as appropriate.
Feel free to use any written resource (notes, books, etc.) but DO NOT
talk to anybody about this exam (other than me).
1. (5 points per part) Sketch (carefully) the following sets. If your sketch
looks ambiguous to you, be sure to describe the set in words, too (for
your own good).
(a) z + z, where z ∈ C
(b) − π2 < Arg(z) < − π4
(c) domain of log(z)
(d) domain of Log(z)
(e) reiθ , with 1 ≤ r ≤ 2 and
π
4
≤θ≤
3π
4
2. (10 points per part)
(a) Explain in your own words the link between the periodicity of ez
and the fact that log(z) is multiple-valued.
(b) Briefly explain, again, in your own words, how to use branch cuts
to create single-valued functions from multiple-valued functions.
3. (15 points) Let C be the semicircle
from 3i to -3i going through 3, and
R
3
let f (z) = z − 1. Evaluate C f (z)dz as a contour integral. Be sure to
indicate z(t) explicitly.
4. (a) (10 points) Determine where the function f (z) = x3 +3xy 2 +i(y 3 +
3x2 y) is differentiable.
(b) (5 points) Determine where f (z) is analytic.
(c) (10 points) Describe a contour to which Cauchy-Goursat applies.
What does Cauchy-Goursat say in this case?
5. (15 points) Find a complex number A such that (2 + 2i)A is real.
1