This exam has a total of 100 points. You have 50 minutes. Partial credit will be awarded so showing your work can only help your grade
Question 1: Restrict
, and take the corresponding branch of the logarithm:
- (a)

- (b)

- (c)

- (d)

Question 2: Compute the following line integrals:
- (a) Let
for
. Compute the line integral

- (b) Let
for
. Compute the line integral

- (c) Let
for
. Compute the line integral

Question 3: Let
, verify that
is harmonic
and find a function
so that
and
is a
holomorphic function.
Question 4: Explain why there is no complex number
so that
.
Question 5: We often use the formulas from ordinary calculus to compute complex derivatives. This problem is part of the justification. Show that if
is holomorphic then
.
Comment: This problem shows that if
and
is a function in the complex plane, and
and
, then we can use this problem to show that
. We will
see later that if two holomorphic functions agree on a line then they agree everywhere. So it would have to be the case that
. (For example, take
and
then
, so it must be that
.)
Question 6: Decide whether or not the following functions are holomorphic where they are defined.
- (a)

- (b)

- (c) Let
and let 
- (d) Let
and let 
- (e) Let
and let 
Question 7: State 4 ways to test if a function
is holomorphic.