Complex Analysis/Sample Midterm Exam 2

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

c. Find all 4 roots of

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.

b. Let and let .
c. where satisfies

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 .


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.