1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 33

Theorem 1. 7 The product of two nonzero complex numbers is a complex

number whose modulus is the product of the moduli of the factors and

whose argument is the sum of their arguments (up to a multiple of 211").

Proof Let z1 = r1 eilli and z2 = r2ei^112 , where 81 and 82 are two particular

values of the arguments of z 1 and z 2 , respectively. Then from property 2

above we have

which shows that

Note

where

In fact, since

it follows that

if - 11" < Arg z1 + Arg z2 ~ 11"

if - 271" < Argz1 + Argz2 ~ -11"

if 11" < Argz 1 + Argz 2 ::=; 271"

-11" < Argz1:::; 11"

\

-11" < Argz2::=;11"

Hence there exists an integer N such that

-11" < Argz1+Argz 2 +2N7!" ~ 11"

which is the same as the integer n( z 1 , z2) given in the statement above.

Theorem 1.8 The quotient of two nonzero complex numbers is a complex
number whose modulus is the quotient of the modulus of the dividend by
the modulus of the divisor and whose argument is the difference between the
argument of the dividend and that of the divisor (up to a multiple of 211").

Proof With the same notations as in Theorem 1. 7, we have, on applying

property 4,

z1 = r1 ei(ll1-ll2)

Z2 r2