The Chemistry Maths Book, Second Edition

8.4 Complex functions 235

Therefore,

cos 15 θ 1 = 1 cos

5

1 θ 1 − 1101 cos

3

1 θ 1 sin

2

1 θ 1 + 151 cos 1 θ 1 sin

4

1 θ 1 = 1161 cos

5

1 θ 1 − 1201 cos

3

1 θ 1 + 151 cos 1 θ

sin 15 θ 1 = 1 sin

5

1 θ 1 − 1101 sin

3

1 θ 1 cos

2

1 θ 1 + 151 sin 1 θ 1 cos

4

1 θ 1 = 1161 sin

5

1 θ 1 − 1201 sin

3

1 θ 1 + 151 sin 1 θ

and the final expressions have been obtained by using the formulasin

2

1 θ 1 + 1 cos

2

1 θ 1 = 11.

0 Exercises 25–27

8.4 Complex functions

Letg(x)andh(x)be (real) functions of the real variable x. A complex function of x

is then

f(x) 1 = 1 g(x) 1 + 1 ih(x) (8.29)

where. Such a function differs in no essential way from a real function of one

variable, except that the possible values of the function are complex numbers. The

discussion of the properties of complex numbers applies to complex functions; for

example, the complex conjugate functionf*(x)is defined by

f*(x) 1 = 1 g(x) 1 − 1 ih(x) (8.30)

with property

f(x)f*(x) 1 = 1 |f(x)|

2

1 = 1 g(x)

2

1 + 1 h(x)

2

(8.31)

where|f(x)|is the modulus of the function. The quantityff*plays an important

role in wave theories, when the wave function is complex.

EXAMPLE 8.8

(i) Express the complex functionf(x) 1 = 12 x

2

1 + 1 (7 1 + 12 i)x 1 − 1 (4 1 + 1 i)in the formf(x) 1 =

g(x) 1 + 1 ih(x), whereg(x)andh(x)are real. (ii) Solveg(x) 1 = 10 ,h(x) 1 = 10 , thenf(x) 1 = 10.

(iii) Find|f(x)|

2

(i)f(x) 1 = 1 (2x

2

1 + 17 x 1 − 1 4) 1 + 1 i(2x 1 − 1 1)

(ii) g(x) 1 = 12 x

2

1 + 17 x 1 − 141 = 1 (2x 1 − 1 1)(x 1 + 1 4) 1 = 10 whenx 1 = 1122 andx 1 = 1 − 4

h(x) 1 = 12 x 1 − 111 = 10 whenx 1 = 1122

Thereforef(x) 1 = 10 whenx 1 = 1122

(iii)|f(x)|

2

1 = 1 g(x)

2

1 + 1 h(x)

2

1 = 14 x

4

1 + 128 x

3

1 + 137 x

2

1 − 160 x 1 + 117

0 Exercise 28

i=− 1