(10 pts) Let γ = [1, 2] + α + [2i, i] − β where α(t) = 2e it, and β(t) = e it for 0 ≤ t ≤ 5π/2. Decide if γ ∼ 0 in the following sets. (a) C∖[0, 1/2]; (b) C∖{2 1/3 + i2 1/3}
In this problem, we shall learn how to compute ∫ 0 + ∞ sint t dt via complex integration. Let us first define I(r) = ∫ |z| = r z −1 e iz dz. (a) (8 pts) Show that I(r) → 0 as r → ∞. (b) (10 pts) Show that I(r) → π i as r → 0. (c) (2 pts) For 0 < s < r < ∞, consider a path γ(s, t) = [s, r] + γr + [−r, − s] − γs where γr(t) = re it and γs(t) = se it on t ∈ [0, π]. What is ∫ γ z −1 e it dz? (d) (10 pts) From (a)-(c), evaluate ∫ 0 + ∞ sint t dt. (Hint: t ↦ sint/t is an even function.)
(10 pts) Let U ⊂ C be an open-set. A function f : U → C satisfies a Lipschitz condition of order α(0 < α ≤ 1) in U if |f(z2) − f(z1)| ≤ m|z2 − z1| α for all points z1, z2 ∈ U, where m is a constant. If f is analytic in U, and an open disk Δ(z0, r) = {z:|z − z0| < r} is contained in U , then prove that |f′(z0)| ≤ mr α − 1 .