MAT 542  Assignment No. 5  Due: March 11, 2002

(1) Evaluate the integral ∫_{γ} \frac{cosz}{(z−π)^{3}} dz in the following cases: a) γ(Θ) = 4e^{ıΘ} , 0 ≤ Θ ≤ 2π.

b) γ(Θ) = 3e^{ıΘ} , 0 ≤ Θ ≤ 2π.

(2) Let f be holomorphic at each point of ℂ and suppose that there is an integer n and two positive real numbers R and M such that |f(z)| ≤ M|z|^{n} for |z| ≥ R. Show that f(z) is a polynomial of degree at most n.

(3) Let f be holomorphic in the disc |z| < R and let M(r) = sup {|f(z)|; |z| = r} for 0 ≤ r < R. Prove:

a) M(r) is a continuous, monotone non-decreasing function of r for 0 ≤ r < R.

b) If f is not constant, M(r) is strictly increasing.

(4) Let f be holomorphic in a bounded domain D, continous in the closure D̄. Suppose that N > 0 is the minimum value of |f(z)| in D̄. Prove that |f(z)| > N for z in D unless f is a constant. Show by example that |f(z)| may attain its minimum value at a point of D if N = 0, without being constant in D.

(5) Let D be a simply connected open subset of ℂ. Let f be a holomorphic function on D. If f(z) ≠ 0, all z ∈ D, there is a holomorphic function F on D such that e^{F (z)} = f(z) all z ∈ D. Prove this theorem.

(6) Let f(z) be holomorphic in 0 < |z| < 1. If |f(z)| ≤ M|z|^{α} for 0 < |z| < 1 (for some M > 0, and some α ∈ ℝ) what can you conclude about the singularity of f at z = 0 ?

(7) Let f be an entire function (holomorphic function on ℂ). In each case, prove or give a counter-example.

a) |f| is constant ⟹ f is constant.

b) lim_{z→∞} f(z) = 0 ⟹ f is constant.

c) If there is a sequence z_{n} with lim_{n→∞} z_{n} = ∞ and lim_{n→∞} f(z_{n}) = 0, then f is constant.

d) Arg f constant ⟹ f is constant.

(8) Let g be holomorphic in 0 < |z| < 1. Prove or give counterexamples.

a) If g has a removable singularity at z = 0, so does e^{g}.

b) If g has a pole at z = 0, so does e^{g}.

c) If g has an essential singularity at z = 0, so does e^{g}.