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HW 1 for Math 419, Fall 2009 1.2/2a: Show that Re(iz) = −Im(z). 1.2/4: Verify that each of the two numbers z = 1 ± i satisfies the equation z 2 − 2z + 2 = 0. 1.3/1bc: Reduce each of these quantities to a real number: 5i ; (c) (1 − i)4 . (b) (1−i)(2−i)(3−i) 1.3/4: Prove that if z1 z2 z3 = 0, then at least one of the three factors is zero. (Hint: Write (z1 z2 )z3 = 0 and use a similar result involving two factors.) 1.4/1ad: Locate the numbers z1 + z2 and z1 − z2 vectorially (is that a word?) when (a) z1 = 2i, z2 = 32 − i; (d) z1 = x1 + iy1 , z2 = x1 − iy1 . 1.4/6b: Give a geometric argument why |z − 1| = |z + i| represents a line through the origin with slope −1. 1.5/2a: Sketch the set of points determined by the condition Re(z̄ − i) = 2. 1.5/10: Prove that (a) z is real if and only if z̄ = z; (b) z is either real or pure imaginary if and only if z̄ 2 = z 2 . 1.8/2: Show that (a) |eiθ | = 1 and (b) eiθ = e−iθ . 1.8/9 (not the bit about Lagrange): Establish the identity 1 + z + z 2 + ... + z m = 1 − z m+1 1−z (z 6= 1). (Hint: Set S to the lefthand side and consider S − zS.) 1