1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 29

correspondence between the complex numbers and the vectors with the

origin 0, provided that we associate the zero complex number (0, 0) to the

zero of null vector 00. This representation of complex numbers by vectors

of origin 0 will be widely used in what follows.

We note that the projections OP' and OP" of the vector OP upon

the axes OX and OY, respectively, have signed measures equal to a and

b, respectively. Denoting the signed measures of OP' and OP", by OP

1

-II

and OP , we have

and op" =b

On the other hand, the length r of the vector OP is given by

r = y' a2 + b2 = lzl (1.7-1)

The measure B in radians of the oriented angle from the positive real axis

to the vector OP is called the argument or the amplitude of the vector

OP, as well as of the corresponding number z -:/-O, and we write

B = argz (1.7-2)

This number B is determined only up to multiples of 27r. Sometimes the

angle itself, also denoted B, is called the argument (or the amplitude) of

OP, or of the number z, but this seldom causes any confusion.

In most cases it is convenient to choose a principal value, or determina-

tion, or branch of the argument. Usually we shall select, as the principal

value of the argument, the value Bo satisfying the inequality -7!" < Bo ::=; 7r,

and write

Bo= Argz ,(1.7-3)

to denote this value. Hence

arg z = Arg z + 2k7r (1.7-4)

where k is an arbitrary integer. However, in some situations it is more

convenient to select the value B 1 satisfying the inequality 0 ::=; Bl < 27r as

the principal value of the argument. In either case the principal value of
the argument of z results in a discontinuous function of z. This is discussed
in more detail in Section 3.12. For the complex number z = 0 the argument
is not defined, so that arg z is not continuous in the whole plane either.
From trigonometry we have, for z -:/-0,

a Rez

cosB = -

; - Tzl'

sinB = ~ =

r

Imz

Tzf

(1.7-5)