1550251515-Classical_Complex_Analysis__Gonzalez_

52 Chapter^1

since the right-hand side means

{ rmfnei(m1J+2krr)/n : k = O, 1, ... , n _ 1}

which coincides with (1.13-2) iff m and n are relatively prime. For instance,

*ifi"4 = *~ = {l,ei"/3,e2irr/3,-l,e4i1r"/3,e5irr/3}

while ( * ifi°)^4 = i^213 has only three values.

Let A, and B be the sets of values represented by the radicals * y'z and

• y'z', respectively. Then we define

(* \(Z)(* \(Z') ={ab:. a EA, b EB} (1.13-5)

We have, in general,

c vz)C \/Z') =1= * itZZ'

since under the interpretation (1.13-5) [which is consistent with (1.13-1)],
the inequality (1.13-4) is just a particular case.
Even if we make use of the principal nth roots, the rule

\/Z \/Z' = ifd (1.13-6)

would not be valid.

Example r-1r-1=1=Jc-1)(-1)=Vi=1

since the left-hand side equals i^2 = -1. Thus the product of two principal

roots is not necessarily equal to the principal root of the product of the
numbers under the radicals.
Multiplication of a radical * y'z = A by a number c E C may be
conveniently defined by

c(*\/Z) ={ca: a EA} (1.13-7)

Addition of two radicals may be defined by a rule analogous to (1.13-5),

namely,

(1.13-8)

Here we have defined addition of two sets of numbers and not ordinary

addition of two numbers. Also, the operation A + B (as defined above)

should not be confused with the union A U B of the two sets. Although

A U A = A for every set A, in general A + A =/: A and also A + A =/: 2A.

Thus, in general, we have

Example

* \(Z + * \(Z =/: 2( * \(Z)

*yC"l + *yC"l = {2i, -2i, O}

(1.13-9)