Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.1 FOURIER TRANSFORMS

and (13.3) becomes

f(t)=

1

2 π

∫∞

−∞

dω eiωt

∫∞

−∞

du f(u)e−iωu. (13.4)

This result is known asFourier’s inversion theorem.

From it we may define theFourier transformoff(t)by

̃f(ω)=√^1

2 π

∫∞

−∞

f(t)e−iωtdt, (13.5)

and its inverse by

f(t)=

1

√

2 π

∫∞

−∞

̃f(ω)eiωtdω. (13.6)

Including the constant 1/

√

2 πin the definition of ̃f(ω) (whose mathematical

existence asT→∞is assumed here without proof) is clearly arbitrary, the only

requirement being that the product of the constants in (13.5) and (13.6) should

equal 1/(2π). Our definition is chosen to be as symmetric as possible.

Find the Fourier transform of the exponential decay functionf(t)=0fort< 0 and

f(t)=Ae−λtfort≥0(λ>0).

Using the definition (13.5) and separating the integral into two parts,

̃f(ω)=√^1

2 π

∫ 0

−∞

(0)e−iωtdt+

A

√

2 π

∫∞

0

e−λte−iωtdt

=0+

A

√

2 π

[

−

e−(λ+iω)t

λ+iω

]∞

0

=

A

√

2 π(λ+iω)

,

which is the required transform. It is clear that the multiplicative constantAdoes not
affect the form of the transform, merely its amplitude. This transform may be verified by
resubstitution of the above result into (13.6) to recoverf(t), but evaluation of the integral
requires the use of complex-variable contour integration (chapter 24).

13.1.1 The uncertainty principle

An important function that appears in many areas of physical science, either

precisely or as an approximation to a physical situation, is theGaussianor

normaldistribution. Its Fourier transform is of importance both in itself and also

because, when interpreted statistically, it readily illustrates a form ofuncertainty

principle.