The Chemistry Maths Book, Second Edition

240 Chapter 8Complex numbers

The coordinates of the rotated point are therefore related to the coordinates of the

unrotated point by

x′ 1 = 1 x 1 cos 1 θ 1 − 1 y 1 sin 1 θ

(8.41)

y′ 1 = 1 x 1 sin 1 θ 1 + 1 y 1 cos 1 θ

The functione

iθ

can be regarded as a representation of the rotation operatorR

θ

which transforms the coordinates(x, y)of a point zinto the coordinates(x′, y′)of

the rotated point z′:

R

θ

(x, y) 1 = 1 (x′, y′) (8.42)

Equations (8.41) play an important role in the mathematical formulation of rotations

(see Chapter 18).

0 Exercise 44

8.6 Periodicity

The trigonometric functionscos 1 θandsin 1 θare periodic functions of θwith period

2 π(see Section 3.2). It follows that the exponential functione

iθ

is also periodic with

period 2 π. Thus, if θis increased by 2 π,

e

i(θ+ 2 π)

1 = 1 e

iθ

1 × 1 e

2 πi

1 = 1 e

iθ

(cos 12 π 1 + 1 i 1 sin 12 π)

Therefore, becausecos 12 π 1 = 11 andsin 12 π 1 = 10 , it follows thate

2 πi

1 = 11 and

e

i(θ+ 2 π)

1 = 1 e

iθ

More generally, the function is unchanged when a multiple of 2 πis added to θ:

e

i(θ+ 2 πn)

1 = 1 e

iθ

, n 1 = 1 0, ±1, ±2,1= (8.43)

Graphically, changing the argumentθby 2 πncorresponds to moving the represen-

tative point on the unit circle through nfull rotations back to its original position

(nanticlockwise rotations if nis positive,|n|clockwise rotations if nis negative).

The function e

iθ

occurs in the physical sciences whenever periodic motion is

described or when a system has periodic structure. We consider here three important

representative classes of physical situations exhibiting periodic behaviour.

Periodicity on a circle. The n nth roots of 1

Figure 8.8 shows three equidistant points on the unit

circle, at the vertices of an equilateral triangle. The

points correspond to complex numbers

z

k

1 = 1 e

(2πk 2 3)i

, k 1 = 1 0, 1 1, 12

with unit modulus and arguments 2 πk 23. These

numbers have the property

()

()

ze e

k

323 ki ki

3

2

= 1

( )

==

ππ

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0

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1

z

2

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Figure 8.8