Preliminary Exam in Complex Analysis
January 2012
Instructions
All assertions require written justification. In particular, state and verify the hypotheses of
any theorems you use. In complex analysis the terms ’analytic’ and ’holomorphic’ are used
interchangeably.
You are free to make use of the following without justification: R∞
−∞ e−x2dx =√π.
1. In the following three part problem, either prove the assertion or give a counterexample.
(a) Suppose Ω is a simply connected region, fis analytic in Ω and f0(z)6= 0 for all z∈Ω.
Then fis one-to-one in Ω.
(b) If fis a non-constant, entire function that satisfies |f(z)| ≤ 2|zez|, then fhas an
essential singularity at infinity.
(c) Let fbe an analytic function in the punctured disk, ∆0={z|0<|z|<1},with a
pole at 0. Then there is a value M > 0,so that for any wwith |w|> M there is a
z∈∆0for which f(z) = w.
2. Find the Laurent expansions of f(z) = z+ 2
z2−z−2in powers of zand 1/z, converging in
the indicated domains
(a) {z|1<|z|<2}
(b) {z|2<|z|<∞}.
3. Let s, R > 0. By integrating f(z) = e−z2over the rectangle with vertices ±R, ±R+is
and letting Rapproach ∞, find the Fourier transform Z∞
−∞
e−iste−t2dt.
4. Suppose that fis analytic in the open unit disk ∆ = {z| |z|<1}, continuous on the
closed unit disk and |f(z)|= 1 for |z|= 1. Show that
(a) fcan be extended to be analytic in C, except for finitely many poles, by defining
f(z) = f1
¯z!−1
in the exterior of the unit disk ∆.
(b) fis rational and has the form f(z) = λz−a1
1−¯a1z. . . z−an
1−¯anz,|ai|<1 and |λ|= 1.