1550251515-Classical_Complex_Analysis__Gonzalez_

28 Chapter 1

z = (a, b) = a + bi is made to correspond to the point P with coordinates

x = a, y = b, and conversely. This procedure establishes a one-to-one

mapping f between the set of all complex numbers and the set of proper

points of the Euclidean plane. This plane, regarded as the image of <C under

f, is called the complex plane (also the z-plane or Gaussian plane).

Example The complex number -2+3i is represented by the point (-2, 3)

(Fig. 1.2)

The point P of the plane corresponding to the complex number z = (a, b)

is sometimes called the affix (Cauchy) or the carrier of this number.

We note that the complex real numbers (a, 0) = a are represented by

points on the OX axis and, conversely, points on this axis correspond to

complex real numbers. Because of this, 0 X is called the real axis, or the

axis of the real numbers. Similarly, the pure imaginary numbers (0, b) are

represented by points on the OY axis, and conversely. Hence OY is called

the imaginary axis, or the axis of the pure imaginary numbers. The origin

0 corresponds to the zero complex number (0, 0).

Because of the one-to-one mapping just established between complex

numbers and points of the plane, the words complex number and point (of

the plane) are often used interchangeably. It is also possible to introduce

a geometric language directly in <C by using the following dictionary:

Point: complex number z = (a, b)

Real axis: the set {(a,O): a ER}

Imaginary axis: the set {(O, b) : b E R}

Origin: the point (0, 0)

Distance between two points: the nonnegative real number Jz1 - z2 J

etc.

Each point P of the complex plane, other than the origin, determines

a directed segment of geometric vector OP (called its position vector),

with origin 0 and endpoint P, and conversely. Hence our correspondence

between complex numbers and points of the plane induces a one-to-one

y

(-2, sir--- 3

I

I

I

I

I

-2 (^0) x
Fig. 1.2