Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.5 COMPLEX LOGARITHMS AND COMPLEX POWERS

To avoid the duplication of solutions, we use the fact that−π<argz≤πand find

z 1 =2^1 /^3 ,

z 2 =2^1 /^3 e^2 πi/^3 =2^1 /^3

(

−

1

2

+

√

3

2

i

)

,

z 3 =2^1 /^3 e−^2 πi/^3 =2^1 /^3

(

−

1

2

−

√

3

2

i

)

.

The complex numbersz 1 ,z 2 andz 3 ,togetherwithz 4 =2i,z 5 =− 2 iandz 6 =1arethe
solutions to the original polynomial equation.
As expected from the fundamental theorem of algebra, we find that the total number
of complex roots (six, in this case) is equal to the largest power ofzin the polynomial.

A useful result is that the roots of a polynomial with real coefficients occur in

conjugate pairs (i.e. ifz 1 is a root, thenz 1 ∗is a second distinct root, unlessz 1 is

real). This may be proved as follows. Let the polynomial equation of whichzis

arootbe

anzn+an− 1 zn−^1 +···+a 1 z+a 0 =0.

Taking the complex conjugate of this equation,

a∗n(z∗)n+a∗n− 1 (z∗)n−^1 +···+a∗ 1 z∗+a∗ 0 =0.

But theanare real, and soz∗satisfies

an(z∗)n+an− 1 (z∗)n−^1 +···+a 1 z∗+a 0 =0,

and is also a root of the original equation.

3.5 Complex logarithms and complex powers

The concept of a complex exponential has already been introduced in section 3.3,

where it was assumed that the definition of an exponential as a series was valid

for complex numbers as well as for real numbers. Similarly we can define the

logarithm of a complex number and we can use complex numbers as exponents.

Let us denote the natural logarithm of a complex numberzbyw=Lnz,where

the notation Ln will be explained shortly. Thus,wmust satisfy

z=ew.

Using (3.20), we see that

z 1 z 2 =ew^1 ew^2 =ew^1 +w^2 ,

and taking logarithms of both sides we find

Ln (z 1 z 2 )=w 1 +w 2 =Lnz 1 +Lnz 2 , (3.34)

which shows that the familiar rule for the logarithm of the product of two real

numbers also holds for complex numbers.