1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 45

are said to be square roots, for n = 3 the roots are said to be cubic roots,

for n = 4 fourth roots, and so on.

Example The cubic roots of 8 are 2, -1 + v'3i, and -1 - v'3i.

When the number w is expressed in binomial form, the exact or ap-

proximate computation of its nth roots is rather difficult, except in the

case n = 2.

Example To find the square roots of -4, we must determine the complex

numbers x + iy satisfying the condition

or

(x+iy)^2 =-4

(x^2 -y^2 ) + 2xyi = -4 + Oi

From the definition of equality it follows that

x2 -y2 = -4 and 2xy = 0

The second equation implies that x = 0 or y = 0. However, y = 0 substi-

tuted into the first equation gives x^2 = -4, which is impossible since the

square of any real number is nonnegative. Hence x = 0, and from the first

equation we get y^2 = 4, so that y = 2 or y = -2. Thus -4 has two square

roots in the complex field: 2i and -2i.

Next consider the general problem of finding the square roots of a+ bi.

We must have

(x + iy)^2 = (x^2 - y^2 ) + 2xyi =a+ bi

Hence we obtain the system

x2 -y2 =a

2xy = b

Squaring both equations and adding, we get

x4 + 2 x2y2 + y4 = a2 + b2

which gives

x2 + y2 = J a2 + b2

(1.12-2)

(1.12-3)

(1.12-4)

where the radical on the right-hand side has the usual interpretation, i.e.,

the principal (nonnegative) square root of a^2 + b2, since x^2 + y^2 2'.: 0.

Letting r = v a^2 + b^2 , we have

x2 + y2 = r (1.12-5)