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