Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

(i) Differentiation:

F

[

f′(t)

]

=iω ̃f(ω). (13.28)

This may be extended to higher derivatives, so that

F

[

f′′(t)

]

=iωF

[

f′(t)

]

=−ω^2 ̃f(ω),

andsoon.

(ii) Integration:

F

[∫t

f(s)ds

]

=

1

iω

̃f(ω)+2πcδ(ω), (13.29)

where the term 2πcδ(ω) represents the Fourier transform of the constant

of integration associated with the indefinite integral.

(iii) Scaling:

F[f(at)]=

1

a

̃f

(ω

a

)

. (13.30)

(iv) Translation:

F[f(t+a)]=eiaω ̃f(ω). (13.31)

(v) Exponential multiplication:

F

[

eαtf(t)

]

= ̃f(ω+iα), (13.32)

whereαmay be real, imaginary or complex.

Prove relation (13.28).

Calculating the Fourier transform off′(t) directly, we obtain

F

[

f′(t)

]

=

1

√

2 π

∫∞

−∞

f′(t)e−iωtdt

=

1

√

2 π

[

e−iωtf(t)

]∞

−∞

+

1

√

2 π

∫∞

−∞

iω e−iωtf(t)dt

=iω ̃f(ω),

iff(t)→0att=±∞,asitmustsince

∫∞

−∞|f(t)|dtis finite.

To illustrate a use and also a proof of (13.32), let us consider an amplitude-

modulated radio wave. Suppose a message to be broadcast is represented byf(t).

The message can be added electronically to a constant signalaof magnitude

such thata+f(t) is never negative, and then the sum can be used to modulate

the amplitude of a carrier signal of frequencyωc. Using a complex exponential

notation, the transmitted amplitude is now

g(t)=A[a+f(t)]eiωct. (13.33)