1.2 Some Geometry 13
An illustration of
the triangle inequality:
lz+wl 5 1:5! + W!
Figure 1.6
If C and 6 are two (other) complex numbers, then by putting z = C — i and
w=§W6g€thi SIC-CI +|Cior
lCl—lél le—él-
Likewise,
lfl—lCl SIC—(El,
which together y
1.3 Subsets of the Plane 27
An open connected set is called a domain. Domains are the natural setting for
the study of analytic and harmonic functions.
A set S is convex if the line segment pq joining each pair of points p, q in S also
lies in S. In parti
1.3 Subsets of the Plane 25
Example 8 The boundary of the set of those 2 with |Im z| > 1 is the set of those 2
with |Im z| = 1. E1
Example 9 The boundary of the set of those 2 = x + i y with x2 < y is the parabola
2
y = x , [:1
Example 10 The boundary of
1.2 Some Geometry 21
In Exercises 1 1 to 17, write the equation of the given circle or straight line in complex
number notation. For example, the circle of radius 4 centered at the point 3 — 2i is
given by the equation |z — (3 — 2i)| = 4.
11.
12.
13.
14.
1.3 Subsets of the Plane 23
Example 1 Each open disc D = {2: |z — zol < R} is an open set. For if r =
|w0 — zol < R, choose a = (R — r)/3. Then, for any 2 with |z — wol < a, we have
|z—zo|<|z—w0|+|wo—zo|<e+r=(R—r)/3+r<R,
by the triangle inequality. Thus,
1.2 Some Geometry 19
of the circle from the family C2 is at the point t = ia (a real), and this circle must
pass through p and — p. Let 2 = x + i y be on both circles (see Fig. 1.10).
ia is center of C
x is center of C2
Figure 1.10
Since 2 is on the cir
lw — cl = plwl-
Upon squaring and transposing terms, this can be written as
|w|2(1— p2) — 2 Re WE + |c|2 = 0.
We complete the square of the left side and ﬁnd that
_ lcl2 lclzp2
1— 2 2 — 2 R =
( p)le ewc+1_p2 1_p
Equivalently,
C P
w—1_p2 =|c|1_p2.
1.2 Some Geometry 15
then (again by Exercise 14, Section 1) we have n0 = II] + 21tj for some integer j. The
values j = 0, . . . , n — 1 yield distinct numbers cos 6]- + i sin 61-, whereas other values
of j just give a repetition of numbers already obtaine
1.1 The Complex Numbers and the Complex Plane 5
z = [2] (c056 + isino)
(b)
Figure 1.3
are two complex numbers. Then
zw = |z| |w|{(cos 6 cos up — sin 6 sin alt) + i(cos 6 sin up + cos up sin 6)}
= |zw|{cos(6 + 11/) + isin(6 + W}.
Moreover,
i _ |z|(cos 6
1.1 The Complex Numbers and the Complex Plane 7
by invoking the formula derived above for the polar representation of the product
of two complex numbers. Thus, if the equality holds for m, then it holds for m + 1.
Since we know it is true for n = 1, it is
1.1 The Complex Numbers and the Complex Plane 9
In summary, the usual xy-plane has a natural interpretation as the location
of the complex variable z = x + i y, and all the rules for the geometry of the vectors
P(x, y) can be recast in terms of z. Hencefo
1.1.1 A Formal View of the Complex Numbers 11
The additive identity is 0 = (0, 0), since (0, 0) + (x, y) = (x, y) + (0, 0) = (x, y) for all
(x, y). The multiplicative identity is 1 = (1, 0), since (1, 0)(x, y) = (x, y)(1, 0) = (x, y)
for all (x, y). Furth
12.
13.
14.
15.
16.
1.3 Subsets of the Plane 29
Which, if any, of the sets given in Exercises 1 to 8 contains 00?
(a) Show that the union of two nonempty open sets is open. Do the same,
replacing “open” with “closed.” Do the same replacing “union” with
“i