1550251515-Classical_Complex_Analysis__Gonzalez_

44

y

0

Fig. 1.12

P4(z4)

I

I

I

I

I

P3(z3)

I

I

I

I

P2(z2)

I I

I

I P1(Z)

Chapter 1

x

This value of the argument of i^3 (i.e., the principal value) is not obtainable

from the second expression for any suitable value of the integer k 2 • The

correct relationship is

arg z n = n arg z + 2k7r

where k is any integer.

Geometrical interpretation. Since for n > 1 the power of a complex

number is a special type of product, it suffices to apply repeatedly the

construction given in Section 1.10, part (c). This is illustrated in Fig. 1.12,

where the vectors OP2, OP 3 , and OP 4 , corresponding to z^2 , z^3 , and

z^4 , have been constructed from the vector OP 1 , corresponding to z. To

construct z-m( m > 0), we note that z-m = ( z-^1 )m, so that we may begin

by constructing 1/ z and proceed as described above.

1.12 EXTRACTION OF ROOTS

Definidon 1.5 Given a complex number wand an integer n > 1, if there

is a complex number z such that

(1.12-1)

it is said that z is an nth root of w.

We shall show that any complex number other than zero has n distinct

nth roots. If w = 0, the only root is z = 0, but we shall consider it as a

multiple root of order n (Sections 5.14 and 5.15). For n = 2 the roots of w