38 CHAP'l'ER l • COMPLEX NUMBERS
(b) Use part (a) and De Moivre's formula to derive Lagrange's identity:
1 +cos 8 +cos 2B + · · · +cos nil = 2 1 + sinf{n+t)6] i. , where 0 < 8 < 21T.
810 ~- lf 1 = zo, z1, ... , Zn-1 are the nth roots of unity, prove that
(z - zi) (z - z2) · · · (z -Z n-1) = 1 + z + z^2 + · · · + zn-^1.
13. Let Zk # 1 be an nth root of unity. Prove that 1 + Zk + z~ + · · · + z;-^1 = O.
- Equation (1-40), De Moivre's formula, can be established without recourse to
properties of the exponential function. Note that this identity is trivially true
forn=l.
(a) Use basic t rigonometric identities to show the identity is valid for n = 2.
(b) Use induction to verify the identity for all positive integers.
(c) How would you verify this identity for all negative integers?- Find all four roots of z^4 + 4 = 0, and use them to demonstrate that z^4 + 4 can be
factored into two quadratics with real coefficients. - Verify that Relation (1-41) is valid.
1.6 The Topology of Complex Numbers
In this section we investigate some basic ideas concerning sets of points in the
plane. The first concept is that of a curve. Intuitively, we think of a curve as
a piece of string placed on a flat surface in some type of meandering pattern.
More formally, we define a curve to be the range of a continuous complex-valued
function z (t) defined on the interval [a, b]. That is, a curve C is the range of
a function given by z (t) = (x (t), y (t)) = x (t) + iy (t), for a S t S b, where
both x (t) and y (t) are continuous real-valued functions. If both x (t) and y (t)
are differentiable, we say that the curve is smooth. A curve for which x (t) and
y (t) are differentiable except for a finite number of points is called piecewise
smooth. We specify a curve C as
C: z (t) = x (t) + iy (t) = (x (t), y (t)), for a St Sb, (1-47)
and say that z (t) is a parametrization for the curve C. Note that, with this
parametrization, we are specifying a direction for the curve C, saying that C is
a curve that goes from the initial point z (a) = (x (a), y (a)) = x (a)+ iy (a)
to the terminal point z (b) = (x (b), y (b)) = x (b) + iy (b). If we had another
function whose range was the same set of points as z (t) but whose initial and
final points were reversed, we would indicate the curve that this function d efines
by-C.
- EXAMPLE 1.22 Find parametrizat ions for C and - C, where C is the