Complex Analysis - Maths KU

16 Chapter 1 Complex Numbers and the Complex Plane
O
r
θ
P(r, θ)
pole polar
axis
Figure 1.6 Polar coordinates
46. Without doing any calculations, explain why the inequalities |Re(z)| ≤|z| and
|Im(z)| ≤|z| hold for all complex numbers z.
47. Show that
(a) |z| = |−z| (b) |z| = |¯z|.
48. For any two complex numbers z1 and z2, show that
|z1 + z2| 2 + |z1 − z2| 2 =2 � |z1| 2 + |z2| 2� .
49. In this problem we will start you out in the proof of the first property |z1z2| =
|z1| |z2| in (3). By the first result in (2) we can write |z1z2| 2 =(z1z2)(z1z2).
Now use the first property in (2) of Section 1.1 to continue the proof.
50. In this problem we guide you through an analytical proof of the triangle inequality
(6).
Since |z1 + z2| and |z1| + |z2| are positive real numbers, we have
|z1 + z2| ≤|z1| + |z2| if and only if |z1 + z2| 2 ≤ (|z1| + |z2|) 2 . Thus, it suffices
to show that |z1 + z2| 2 ≤ (|z1| + |z2|) 2 .
(a) Explain why |z1 + z2| 2 = |z1| 2 + 2Re(z1¯z2)+|z2| 2 .
(b) Explain why (|z1| + |z2|) 2 = |z1| 2 +2|z1¯z2| + |z2| 2 .
(c) Use parts (a) and (b) along with the results in Problem 46 to derive (6).
1.3 Polar Form of Complex Numbers
Recall from calculus 1.3 that a point P in the plane whose rectangular coordinates are
(x, y) can also be described in terms of polar coordinates.The polar coordinate system,
invented by Isaac Newton, consists of point O called the pole and the horizontal half-line
emanating from the pole called the polar axis.If r is a directed distance from the pole to
P and θ is an angle of inclination (in radians) measured from the polar axis to the line OP,
then the point can be described by the ordered pair (r, θ), called the polar coordinates of
P .See Figure 1.6.
Polar Form Suppose, as shown in Figure 1.7, that a polar coordinate
system is superimposed on the complex plane with the polar axis coinciding
with the positive x-axis and the pole O at the origin.Then x, y, r and θ are
related by x = r cos θ, y = r sin θ.These equations enable us to express a
nonzero complex number z = x + iy as z =(r cos θ)+ i(r sin θ) or
z = r (cos θ + i sin θ). (1)
We say that (1) is the polar form or polar representation of the complex
number z.Again, from Figure 1.7 we see that the coordinate r can be interpreted
as the distance from the origin to the point (x, y).In other words, we
shall adopt the convention that r is never negative † so that we can take r to
† In general, in the polar description (r, θ) of a point P in the Cartesian plane, we can
have r ≥ 0orr