HW 2 for Math 419, Fall 2009
Due Friday, September 4
8/5d: By writing the individual factors on the left in exponential form,
performing the needed operations,
and finally changing
back to rectangular
√
√ −10
−11
= 2 (−1 + 3i).
coordinates, show that (1 + 3i)
8/10a: Use de Moivre’s formula to derive the trig identity cos 3θ = cos3 θ −
3 cos θ sin2 θ.
10/6:√ Find the four zeros of the polynomial z 4 + 4, one of them being
z0 = 2eiπ/4 = 1 + i. Then use those zeros to factor z 4 + 4 into quadratic
factors with real coefficients. (Hint: The answer is (z 2 +2z +2)(z 2 −2z +2).)
10/7: Show that if c is any nth root of unity other than unity itself, then
1 + c + c2 + . . . + cn−1 = 0. (Hint: Use the first part of 8/9 from hw 1.)
11/1: Sketch the following sets and determine which are domains:
(a) |z − 2 + i| ≤ 1;
(b) |2z + 3| > 4;
(c) Im(z) > 1;
(d) Im(z) = 1;
π
(e) 0 ≤ arg(z) ≤ 4 (z 6= 0); (f) |z − 4| ≥ |z|.
(Hint: (b) and (c) are domains.)
11/2: Which sets in Exercise 1 (the previous one) are neither open nor
closed? (Hint: (e).)
11/3: Which sets in Exercise 1 are bounded? (Hint: (a).)
11/9: Show that any point z0 of a domain is an accumulation point of that
domain.
11/10: Prove that a finite set of points z1 , z2 , . . . , zn cannot have any accumulation points.
12/1: For each of the functions below, describe the domain of definition
that is understood:
(b) f (z) = Arg( 1z );
(a) f (z) = z 21+1 ;
z
1
(c) f (z) = z+z
;
(d) f (z) = 1−|z|
2.
(Hint: (a) z 6= ±i, (c) Re(z) 6= 0.)
12/3: Suppose that f (z) = x2 − y 2 − 2y + i(2x − 2xy), where z = x + iy.
Use the expressions
z+z
z−z
x=
and y =
2
2i
to write f (z) in terms of z, and simplify the result. (Hint: f (z) = z 2 + 2iz.)