The Complex Plane 185Writing
z = |z|e(ArSW+2-fc)\we find
zlM = |z|l/mc(ArgW/m+2Wfc/m)i) fc = 0,...,m-l. (4.1.5)Thus, the m-th roots of a complex number have exactly m distinct values.
On the complex plane, these values are at the vertices of a regular m-gon.
FOR EXAMPLE, taking z = 1 and m = 3 so that Arg(z) = 0, we find the
three values of \/l: 1, (-1 + iy/3)/2, (-1 - iy/Z)/2.
EXERCISE 4.1.5. (a)c Find all the values of \/&i and \/64i and draw them
in the complex plane. (b)A Find all the values of y 1 + iy/3 + y 1 — iy/i.
Hint: one of them is \/6-' simply square the expression.
Another application of (4.1.4) is deriving certain trigonometric identi-
ties. By taking the n-th power on both sides, we get de Moivre' s formula:
(cos# + ism8)n = cosn8 + isinn8; (4.1.6)of course, the French mathematician ABRAHAM DE MOIVRE (1667-1754),
who published (4.1.6) in 1722, did not use (4.1.4) in his derivations.
By equating the real and imaginary parts of (4.1.6), we get the expres-
sions for cos nQ and sin nO in terms of the products of sin 6 and cos 6. FOR
EXAMPLE, n = 2 yields the familiar results: cos 28 = cos^28 — sin^2 8, sin 28 =
2 sin 6 cos 8.
EXERCISE 4.1.6.B Find similar expressions for cos3$ and sin 3^.
Yet another application of (4.1.4) is illustrated by the following exercise.
EXERCISE 4.1.7.c Evaluate the indefinite integral f e~3x cos 2x dx without
integration by parts. Hint: note that the integrand is di(e<-~3+2^x). Integrate
the complex exponential as if it were real, and them compute the real part of the
result.
We conclude this section with a few definitions related to sets in the
complex plane; these definitions are identical to those on page 121. As
with ordinary points in the plane, \z\ — z<z\ is the distance between z\ and
Z2, and the set {z : \z—ZQ\ < r} is an open disk with center at z 0 and radius
r > 0; similarly, {z :\z — ZQ\ < r} is a closed disk. A neighborhood of a
point z is an open disk centered at z. A point in a set is called interior if