1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

1.4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 23

• EXAMPLE 1.7 If z = 1 + i, then r = J2 and z = ( J2cos f, J2sin f) =

J2(cos f + isin ~) is a polar representation of z. The polar coordinates in this

case are r = J2, and 0 = f.

As Figure 1.11 (b) shows, 0 can be any value Tor which the identities cos 0 = ~

and sin 0 = ~ hold. For z =f 0, the collection of all values of 0 for which

z = r(cos O+isin 0) is denoted arg z. Formally, we have the following definitions.

I Definit ion 1.10: arg z

If z =f 0,

arg z = {8: z = r(cos8 + isinO)}.

If 8 E arg z, we say that 0 is an argument of z.

(1-28)

Note thl'lt we write 8 E arg z as opposed to 0 = arg z. We do so because

arg z is a set, and the designation 0 E arg z indicates that 0 belongs to that set.

Note also that, if 01 E arg z and 02 E arg z, then there exists some integer n

such that

(1-29)

• EXAMPLE 1.8 Because 1 + i = J2(cos f + isin f ), we have

arg (I + i ") = -{ 7r 4 +^2 rm: n is. an mteger. } = { · · · , --77r 4'4'4'4' - -7r 97r -l 77r · · · }.

Mathematicians have agreed to single out a special choice of 0 E arg z. It is

that value of 0 for which -7r < 8 ~ 7r, as the following definition indicates.

I Definition 1.11: Arg z

Let z =f 0 be a complex number. Then

Argz = 0, provided z = r (cos8 + isin8) and - 7r < 8 ~ 7r. (1-30)

If 0 = Arg z, we call 0 the argument of z,