Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment III.

1. Problem 1.

a) Prove that

\frac{∂}{∂z̄}(f g) = \frac{∂f}{∂z̄}g + \frac{∂g}{∂z̄}f, \frac{∂}{∂z} (f g) = \frac{∂f}{∂z} g + \frac{∂g}{∂z} f.

b) Let f(z) and zf(z) be two complex-valued harmonic functions (their real and imaginary parts both satisfy the Laplace equation). Prove that f is analytic.

2. Problem 2.

a) Is e^{−1/|z|} uniformly continuous in 0 < |z| ≤ 1?

b) Let f(z) be uniformly continuous in |z| < 1. Prove that for any point ξ on the circle |z| = 1, and for any sequence z_{n} → ξ, |z_{n}| < 1, there exists a limit lim_{n→∞} f(z_{n}).

c) Prove also that the limit does not depend on the choice of the sequence {z_{n}}, and that the function f, defined on the boundary as f(ξ) = lim_{n→∞} f(z_{n}), is continuous in the closed disc |z| ≤ 1.

3. Problem 3.

a) Use the method described in class to develop

\frac{z^{4}}{z^{3} − 1}

into partial fractions.

b) If R(z) is a rational function of order n, how large and how small can the order of R′(z) be?

4. Problem 4.

a) If Q is a polynomial with distinct roots α_{1}, ..., α_{n}, and if P is a polynomial of degree < n, show that

\frac{P (z)}{Q(z)} = ∑_{k=1}^{n} \frac{P (α_{k})}{Q′(α_{k})(z − α_{k})} .

b) Use the above formula to prove that there exists a unique polynomial P of degree < n with given values c_{k} at the points α_{k} (Lagrange’s interpolation polynomial).

5. Problem 5. If a rational function is real on |z| = 1, how are the zeros and the poles situated?