Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

Rez

Imz

z=x+iy

x

y

−y z∗=x−iy

Figure 3.6 The complex conjugate as a mirror image in the real axis.

Inthecasewherezcan be written in the formx+iyit is easily verified, by

direct multiplication of the components, that the productzz∗gives a real result:

zz∗=(x+iy)(x−iy)=x^2 −ixy+ixy−i^2 y^2 =x^2 +y^2 =|z|^2.

Complex conjugation corresponds to a reflection ofzin the real axis of the

Argand diagram, as may be seen in figure 3.6.

Find the complex conjugate ofz=a+2i+3ib.

The complex number is written in the standard form

z=a+i(2+3b);

then, replacingiby−i,weobtain

z∗=a−i(2+3b).

In some cases, however, it may not be simple to rearrange the expression for

zinto the standard formx+iy. Nevertheless, given two complex numbers,z 1

andz 2 , it is straightforward to show that the complex conjugate of their sum

(or difference) is equal to the sum (or difference) of their complex conjugates, i.e.

(z 1 ±z 2 )∗=z∗ 1 ±z 2 ∗. Similarly, it may be shown that the complex conjugate of the

product (or quotient) ofz 1 andz 2 is equal to the product (or quotient) of their

complex conjugates, i.e. (z 1 z 2 )∗=z∗ 1 z∗ 2 and (z 1 /z 2 )∗=z∗ 1 /z∗ 2.

Using these results, it can be deduced that, no matter how complicated the

expression, its complex conjugate mayalwaysbe found by replacing everyiby

−i. To apply this rule, however, we must always ensure that all complex parts are

first written out in full, so that noi’s are hidden.