Math 115 Yum Tong-Siu, Exercises of Mathematics

Download Math 115 Yum Tong-Siu and more Exercises Mathematics in PDF only on Docsity! Math 115 (2006-2007) Yum-Tong Siu 1 Homework Assigned on September 21, 2006 due September 28, 2006 Problem 1 (from Ahlfors p.2, #1 and p.4, #1). Compute the real part and the imaginary part of the following two complex numbers in terms of rational numbers and using the process of taking the square roots of positive numbers. (a) ( 2 + i 3 − 2i )2 . (b) √ 1 − i √ 3 2 . Problem 2 (from Ahlfors p.6, #1). Show that the system of all matrices of the special form ( α β −β α ) , combined by matrix addition and matrix multiplication, is isomorphic to the field of complex numbers. Problem 3 (from Ahlfors p.11, #1). Prove that ∣ ∣ ∣ ∣ a − b 1 − āb ∣ ∣ ∣ ∣ < 1 if |a| < 1 and |b| < 1. Problem 4 (from Ahlfors p.11, #3). If |ai| < 1 and λi ≥ 0 for i = 1, · · · , n and λ1 + λ2 + · · · + λn = 1, show that |λ1a1 + λ2a2 + · · · + λnan| < 1. Problem 5 (from Ahlfors p.15, #2). Prove that the points a1, a2, a3 of an equilateral triangle if and only if a2 1 + a2 2 + a2 3 = a1a2 + a2a3 + a3a1. Problem 6 (from Ahlfors p.16, #1). (a) Express cos 3ϕ, cos 4ϕ, and sin 5ϕ in terms of cos ϕ and sin ϕ and (b) express tan 7ϕ in terms of tan ϕ.