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.