Prelim in Complex Analysis, August 2003

Let C be the complex plane and D = {z ∈ C : |z| < 1} be the open unit disk.

1. Evaluate the following integrals.

∫_{|z|=2} csc z dz,  ∫_{|z|=1} \frac{1 − cos z}{z^{2}} dz,  ∫_{0}^{π} \frac{dθ}{2 + cos θ} .

2. If f(z) is entire and |f(z)| ≤ |z|^{3/2} for all z, show that f is identically zero.

3. Show that a function f : D → C is constant if and only if both f and \overline{f} are analytic in D.

4. Show that the class X of analytic functions f in D with

∫_{D} |f(z)| dx dy ≤ 1

is a normal family.

5. Find the real and imaginary parts of the complex number

z = Log (1 + i) + cos(1 + i),

where Log is the branch of the logarithm on C − {x : x ≤ 0} with Log (1) = 2πi.

6. If f : D → C is a bounded analytic function, show that

sup_{z∈D}(1 − |z|^{2})|f′(z)| ≤ sup_{z∈D} |f(z)|.

7. Show that if f : D → C is analytic and one-to-one, then f′(z) ≠ 0 for every z ∈ D.

8. If f : D → D is analytic and f(0) = f′(0) = 0, show that |f(z)| ≤ |z|^{2} for all z ∈ D.

9. Find the Laurent series of f(z) = 1/[z(1 − z)] at z = 0, at z = 1, at z = 2, and at z = ∞.

10. Let

B(z) = ∏_{k=1}^{n} \frac{a_{k} − z}{1 − \overline{a}_{k}z} ,

where a_{1}, ⋯, a_{n} are distinct points in D − {0}. Show that

B(z) = ∏_{k=1}^{n} \frac{1}{\overline{a}_{k}} + ∑_{k=1}^{n} \frac{1}{\overline{a}_{k}\overline{B′(a_{k})}(1 − \overline{a}_{k}z)}.