Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment XI.

1. Problem 1.

a) Compute

∫_{|z|=r} xdz

by use of a parameter.

b) Compute the integral in a) by observing that x = (z + z̄)/2 = (z + r^{2}/z)/2 on the circle.

c) Compute

∫_{|z|=2} \frac{dz}{z^{2} − 1}

for the positive sense of the circle.

d) Compute

∫_{|z|=1} |z − 1||dz|

2. Problem 2. a) Suppose that f(z) is analytic on a closed curve γ (i. e., f is analytic in a region that contains γ). Show that

∫_{γ} \overline{f(z)}f′(z)dz

is purely imaginary. (The continuity of f′(z) is taken for granted).

b) Assume that f(z) is analytic and satisfies the inequality |f(z) − 1| < 1 in a region Ω. Show that

∫_{γ} \frac{f′(z)}{f(z)}dz = 0

for every closed curve in Ω. (The continuity of f′(z) is taken for granted).

c) If P (z) is a polynomial and C denotes the circle |z − a| = R, what is the value of ∫_{C} P (z)\overline{dz}? Hint: −2πiR^{2}P′(a).

3. Problem 3.

a) Let f(z) be continuous in the domain Ω = {z ∈ C : |z| ≥ R_{0}, ℑz ≥ a}, (a is a fixed real number ), and in this domain f(z) → 0 uniformly on arcs {z ∈ Ω, |z| = R}, as z → ∞. Prove that for any m > 0,

lim_{R→∞} ∫_{Γ_{R}} e^{imz}f(z)dz = 0,

where Γ_{R} is an arc of the circle |z| = R in z ∈ Ω.

Hint: On the half-arc |z| = R, ℑz > 0, use sin θ ≥ 2θ/π, 0 ≤ θ ≤ π/2. In the case a < 0 use the fact that the length of the corresponding arcs tends to |a| as R → ∞.

b) Let f(z) be continuous in the half-plane ℜz ≥ σ, (σ is a fixed real number ), and in this half-plane f(z) → 0 uniformly on arcs {ℜz ≥ σ, |z| = R}, as z → ∞. Prove that for any t < 0,

lim_{R→∞} ∫_{Γ_{R}} e^{zt}f(z)dz = 0,

where Γ_{R} is an arc of the circle |z| = R, ℜz ≥ σ. If f(z) is continuous in ℜz ≤ σ, then the statement is true, provided t > 0, and Γ_{R} is an arc of the circle |z| = R, ℜz ≤ σ.