Complex Analysis/Sample Midterm Exam 1

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 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.

(c) Let and let
(d) Let and let
(e) Let and let

Question 7: State 4 ways to test if a function is holomorphic.