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.)
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