# Complex Analysis/Sample Midterm Exam 2

1. Restrict , and take the corresponding branch of the logarithm:

- a.
- b.
- c. Find all 4 roots of
- d

2. State the Cauchy-Riemann equations for a complex valued function . If you use symbols other then and indicate how they relate to these quantities.

3. State whether the give function is holomorphic on the set where it is defined.

- a.
- b. Let and let .
- c. where satisfies
- d.

4. Let be a simple closed curve so that lies in the interior of the region bounded by .

- a. Suppose and compute
- simply writing the correct value without any explanation will not receive credit.
- b. We now consider the case corresponding to . Please compute
- and explain your steps.
- c: Now suppose and compute

5. Let be given by . Calculate

6. Let find a function so that is holomorphic in the complex plane and .

7.

- a. Using the limit characterization of the complex derivative show that is not holomorphic.
- b. On the other hand show that if .
- c. Do parts (a) and (b) contradict each other, explain why or why not.

8. State Cauchy's integral theorem, and intuitively what you need to know about the function, domain and contour.