Math 419 Exam 1, Fall 2009
This test is worth 100 points. Please provide 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.
1. (3 points per part) Sketch (carefully) the following sets. If your sketch looks ambiguous,
be sure to describe the set in words, too.
(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) Evaluate ii . Please clearly indicate x and y (if you choose rectangular form)
or r and θ (if you choose polar form).
3. (15 points) Let
R C be the semicircle from 2 to -2 going through 2i, and let f (z) =
2
z . Evaluate C f (z)dz as a contour integral and express the solution in rectangular
coordinates.
4. (a) (10 points) Determine the set of points at which the function f (z) = x3 + 3xy 2 +
i(y 3 + 3x2 y) is differentiable
(b) (5 points) Plot the set from part (a).
(c) (5 points) Determine where f (z) is analytic.
5. (5 points per part)
(a) Explain the link between the periodicity of ez (what is the period?) and the fact
that log(z) is multiple-valued. I am not asking for a proof or any computation.
Please use no more than a few sentences at most.
(b) Explain how to use branch cuts to create single-valued functions from multiplevalued functions. Again, I am not asking for proof or computation, and, again,
please use no more than a few sentences at most.
6. Based on our experience over R, it may seem strange that f (z) = |z|2 = x2 + y 2 (where
z = x + iy, as usual) is differentiable only at 0, as we saw in class. We are going to
think through why that is true with this problem.
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(a) (12 points) Draw the images of the circles of radius 3, 2, 1, and 21 in a single
mapping, i.e., two complex planes beside each other, the one on the left being the
domain (where you draw the four circles drawn) and the one on the right being
the codomain.
(b) (6 points) What is the range of f (z), i.e., which points in C are hit by f (z)?
(c) (6 points) How many points are in the preimage of each point in the range of
f (z)?
(d) (3 points) Let’s make a 3-dimensional graph of f (z). In particular, on a 3dimensional coordinate system labelled (x, y, z) with vertical axis z (as usual),
let (x, y) be the complex input and let z be the real output. Draw the circles of
part (a) in the (x, y) plane and plot the image of each point directly above it.
Connect the images of these four circles to get a nice graph of the real part of the
function.
(e) (3 points) Im(f (z)) = 0, so a plot of the imaginary part in 3-space (as in the
previous part) would just be the (x, y) plane. All tangent planes to this imaginary
part are then just the (x, y) plane again. For which point(s) in the previous part
is the tangent plane the (x, y) plane? Since these two tangent planes must be
equal for f (z) to be differentiable (according to the Cauchy-Riemann equations),
at which point(s) is f (z) differentiable?
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