Masters Exam : Complex Analysis

January 1991

Instructions: Do any 5 of the following problems.

1. Consider the many valued function f(z) = (iz + 1)^{1/2}. Choose a branch of this function so that it becomes single valued and analytic and sketch its domain. For this branch, what is f′(z)?

2. Determine whether the following function is entire or not. If so, compute its derivative. f(z) = e^{−y}((x + 1) cos(x) − y sin(x)) + ie^{−y}((x + 1) sin(x) + y cos(x))

3. Let C be the contour C = {z : |z| = 2} with its positive orientation (i.e. counter clockwise). Compute the following integrals.

(a) ∫_{C} \frac{z + 1}{cos(z)} dz  (b) ∫_{C} \frac{cos(z)}{z^{2}} dz  (c) ∫_{C} \frac{sinh(z)}{z^{6}} dz

4. State and prove Liouville’s Theorem.

5. Let f(z) = z^{4}(e^{1/z} − 1). Find all singularities of f(z) and write down the principal part of f(z) at each singular point. Determine whether each singularity is removable, essential, or a pole. Give the residue at each singularity.

6. A non-singular matrix T = ( \begin{matrix}a & b \\ c & d \end{matrix} ), a, b, c, d, ∈ ℝ, defines a linear fractional transfor-

mation by T (z) = \frac{az + b}{cz + d}. Let ℌ = \{z : ℑ(z) > 0\} be the upper half plane. Show that the linear fractional transformation defined by the matrix T preserves ℌ iff det(T) > 0. For which matrices T does the linear fractional transformation satisfy T (z) = z for all z ∈ ℌ?

7. Compute one of the following two integrals.

(a) ∫_{0}^{∞} \frac{dx}{x^{1/2}(x^{2} + 1)}  (b) ∫_{0}^{∞} \frac{x sin(x)}{(x^{2} + 4)(x^{4} + 1)} dx

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