Math 185 Complex Analysis
Fall 2018, Professor Klass
Midterm
Oct 5, 2018
Name:
Student ID:
Instructions
1. This exam contains 5 regular questions: each worth 25 points. Choose four questions
that you want graders to grade. Please specify your choice in the second column of the
following box. Otherwise, only the first 4 will be graded.
2. Please begin by writing your name on the first page. You are allowed 55 minutes for this
exam. These questions are not necessarily ordered in the increasing difficulty. Please
first pick the problems you feel most comfortable with and pace yourself accordingly.
3. You may not use any calculator, notes, or other assistance on this exam. In order to
receive full credit, you must show your work and carefully justify your answers. Partial
credit will be given if you show sufficient progress in solving a problem. If you need
more room, use the backs of the pages and indicate that you have done so. Please
write neatly.
Grade Table
Question Which to Grade
1
2
3
4
5
Total:
Score
Math 185 Complex Analysis
Midterm - Page 2 of 6
1. (a) (12.5 points) Find all the values of ii .
(b) (12.5 points) Simplify
(1 − i)5
√
(1 + i 3)8
into the form a + bi, where a, b ∈ R.
Oct 5, 2018
Math 185 Complex Analysis
Midterm - Page 3 of 6
Oct 5, 2018
2. (a) (18 points) Calculate the following integral
∫
dz
,
2
γ z +z −2
where γ is the circle centered at the origin with radius 3 traversed once in the counterclockwise direction.
(b) (7 points) Please briefly describe another way to calculate the integral above. (We
currently know there are at least three ways.)
Math 185 Complex Analysis
Midterm - Page 4 of 6
Oct 5, 2018
3. (a) (10 points) Please state the Cauchy-Riemann equations.
(b) (15 points) Let f be analytic on a connected open region A. Suppose that |f (z)|
is constant on A. Prove that f (z) is constant on A. (Hint: consider |f (z)|2 .)
Math 185 Complex Analysis
Midterm - Page 5 of 6
Oct 5, 2018
4. (a) (10 points) Please state Liouville’s Theorem.
(b) (15 points) Suppose that f : C → C is non-constant and entire. Prove that for any
w0 ∈ C and any ϵ > 0, there exists z0 ∈ C such that |f (z0 ) − w0 | < ϵ.
Math 185 Complex Analysis
5. (25 points) Please evaluate
Midterm - Page 6 of 6
∫
0
2π
e−iθ ee dθ.
iθ
Oct 5, 2018