Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.1 FOURIER TRANSFORMS

obtained simply by noting from (13.42) that the cross-correlation (13.40) of two

functionsfandgcanbewrittenas

C(z)=

∫∞

−∞

f∗(x)g(x+z)dx=

∫∞

−∞

[ ̃f(k)]∗ ̃g(k)eikzdk. (13.44)

Then, settingz= 0 gives the multiplication theorem
∫∞

−∞

f∗(x)g(x)dx=

∫

[ ̃f(k)]∗ ̃g(k)dk. (13.45)

Specialising further, by lettingg=f, we derive the most common form of

Parseval’s theorem,
∫∞

−∞

|f(x)|^2 dx=

∫∞

−∞

| ̃f(k)|^2 dk. (13.46)

Whenfis a physical amplitude these integrals relate to the total intensity involved

in some physical process. We have already met a form of Parseval’s theorem for

Fourier series in chapter 12; it is in fact a special case of (13.46).

The displacement of a damped harmonic oscillator as a function of time is given by

f(t)=

{

0 fort< 0 ,

e−t/τsinω 0 t fort≥ 0.

Find the Fourier transform of this function and so give a physical interpretation of Parseval’s

theorem.

Using the usual definition for the Fourier transform we find

̃f(ω)=

∫ 0

−∞

0 ×e−iωtdt+

∫∞

0

e−t/τsinω 0 te−iωtdt.

Writing sinω 0 tas (eiω^0 t−e−iω^0 t)/ 2 iwe obtain

̃f(ω)=0+^1

2 i

∫∞

0

[

e−it(ω−ω^0 −i/τ)−e−it(ω+ω^0 −i/τ)

]

dt

=

1

2

[

1

ω+ω 0 −i/τ

−

1

ω−ω 0 −i/τ

]

,

which is the required Fourier transform. The physical interpretation of| ̃f(ω)|^2 is the energy
content per unit frequency interval (i.e. theenergy spectrum) whilst|f(t)|^2 is proportional to
the sum of the kinetic and potential energies of the oscillator. Hence (to within a constant)
Parseval’s theorem shows the equivalence of thesetwo alternative specifications for the
total energy.

13.1.10 Fourier transforms in higher dimensions

The concept of the Fourier transform can be extended naturally to more than

one dimension. For instance we may wish to find the spatial Fourier transform of