1550251515-Classical_Complex_Analysis__Gonzalez_

10 Chapter 1

in (1.1-7), (c,d) is the dividend and (a,b) the divisor. The opera-

tions involved in (1.1-3) g,nd (1.1-7) are called subtraction and division,

respectively.

If in (1.1-6) we assum~ that c = a, d = b we obtain u = 1 and v =

0. Hence the complex number (1, 0) is the neutral element or unity with
   respect to multiplicq,tion; j,!')., it is the only complex number satisfying
   (a, b)(u., v) =(a, b), (a,b):f=(O,O)
   Next, letting c = 1, d = 0 in (1.1-6), we find that u = a/(a^2 + b^2 ) and
   v = -b/(a^2 + b^2 ). H~rn~~

(l, 0) ( a -b )

(a; b) = a^2 + b^2 ' a^2 + b2

This complex number ii:i i:iaid to be the reciprocal of (a, b), denoted (a, b )-^1.

Every complex number <;lXcept (0, 0) has a reciprocal.

The quotient ( c, d) / ( q, b) can be obtained by multiplying ( c, d) by the

reciprocal of (a, b ). In fl'l,ct,

( c, d)( a, b )~1 = ( c, d) ( a2 : b2 ' a2-: b2 )

= ( ac + bd. ad - be )

a2 + b2 ' a2 + b2

As in every field, it follows from the preceding property that the product

of two complex numbers q1:1,nnot be zero unless at least one of. the factors

is zero. To show this, WtJ first note that

(a, b)(O, 0) = (0, 0)

i.e., the product is zero i.f pne factor is the zero complex number. Now

suppose that

(a,b)(c,d)=(O,O) (1.1-8)

and that (a,b) :f: (0,0). J'hen (a,b) has a reciprocal (a,b)-1, and

multiplying both sides of (1.1-8) by (a, b )-^1 , we get ,

(a, b)-^1 [(a, b)(c, d)] =(a, b)-^1 (0,0)

or

[(a, b)-^1 (a, b)]( c, d) = ( c, d) = (0, 0)

If n is a positive integer, the power zn is defined inductively as follows:

z^1 = z and zn = zn-l · z (n > 1). For z :f: 0 we further define z^0 = 1

and z-n = 1/zn.

In Section 1.11 we discuss powers of complex numbers in detail.