Master’s Comprehensive Exam Complex Variables January 1997

1. (10 pts.) Sketch the set of all the points z satisfying the equation:

|2\overline{z} + 4i| = 6

2. (10 pts.) Determine whether or not there exists an entire function f(z) satisfying the following

conditions:

|f(0)| = 1and |f(z)| = 1/2for all zwith |z| = 1

If there is, give an example. If not, explain why.

3. (40 pts.; 10 pts. per each part) Let C be the circle of radius 4 centered at the origin (with the counter-clockwise direction). Compute the following integrals.

(a) ∫_{C} \overline{z}dz;

(b) ∫_{C} \frac{cosh(z)}{(z − 5)(z + 6i)} dz;

(c) ∫_{C} \frac{z^{4}}{(z − i)^{3}} dz;

(d) ∫_{C} \frac{e^{z}}{(z − i)(z − 2i)} dz.

4. (30 pts.; 10 pts. per each part) Find the Laurent series of the function

f(z) = \frac{1}{(z − 1)(z − 2)}

in the following regions:

(a) |z| < 1;

(b) 1 < |z| < 2;

(c) |z| > 2.

5. (10 pts.) Find the linear fractional transformation T which maps the points −1, 0, 1 onto −i, 1, i respectively.