Complex Analysis Preliminary Examination June 1995

U = \{z : |z| < 1\}, \overline{U} = \{z : |z| ≤ 1\}, Γ = \{|z| = 1\}

1. A complex-valued function is said to be harmonic if both its real and imaginary parts are harmonic. Suppose both f(z) and z f(z) are complex-valued harmonic functions on a region R. Show that f(z) is analytic on R.

2. Suppose f is entire and the range of f fails to meet some circle. What can be said about f? Justify your answer.

3. Present a contour integration argument to show that

∫_{−∞}^{∞} \frac{cos x}{(x^{2} + 1)^{2}} dx = \frac{π}{e}

4. Let f be holomorphic in a neighborhood of \overline{U}. Let f(z) = ∑_{k=0}^{∞} a_{k} z^{k} and s_{n}(z) =

∑_{k=0}^{n} a_{k} z^{k}. Show that, among all polynomials P of degree n or less, the integral

\frac{1}{2π} ∫_{0}^{2π} |f(e^{iθ}) − p(e^{iθ})|^{2} dθ attains its minimum for p = s_{n}.

5. A. State Cauchy’s integral formula for an open disk.

B. Let f be in the disk algebra, i.e., let f be continuous on \overline{U} and holomorphic on U. Use part (A) to prove that

f(z) = \frac{1}{2πi} ∫_{Γ} \frac{f(w)}{w − z} dw for z ∈ U .

6. Let f be holomorphic in a neighborhood of 0 and suppose that the series ∑_{k=0}^{∞} f^{(k)}(0) converges. Show that f can be extended to an entire function.

7. Let f be meromorphic in a neighborhood of \overline{U} with no poles on Γ. Let |A| > max_{z∈Γ} |f(z)|. Prove that in U, counting multiplicity and order, the number of solutions to f(z) = A equals the number of poles of f.

8. Let F denote the family of functions f(z) = ∑_{n=0}^{∞} a_{n}z^{n} which are holomorphic in U and satisfy |a_{n}| ≤ n for n = 0, 1, 2, …. Prove that any sequence in F has a subsequence which converges uniformly on compact subsets of U.

9. Let f(z) be nonconstant and holomorphic in a neighborhood of \overline{U} with f(0) = a_{0}. Let M = max_{z∈Γ} |f(z)|. Let λ ∈ U and suppose f(λ) = 0. Prove that |λ| ≥ \frac{|a_{0}|}{M}. [Hint: Consider g(z) = f(\frac{z + λ}{1 + \overline{λ} z} ).]