E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 11.9, pgs. 518-528
Lecture: Fourier Transform Module: 12
Suggested Problem Set: {2, 3, 9, 14(a)}March 31, 2009
Quote of Lecture 12
Jenny said when she was just five years old there was nothin’ happenin’ at all. Every time
she puts on a radio there was nothin’ goin’ down at all, not at all. Then one fine mornin’
she puts on a New York station you know, she don’t believe what she heard at all. She
started shakin’ to that fine fine music you know her life was saved by rock ’n’ roll. Despite
all the amputations you know you could just go out and dance to the rock ’n’ roll station
and it was alright.
The Velvet Underground : Rock And Roll (1970)
We are finally at the end of our study of Fourier methods. We have Fourier series to represent periodic
functions and Fourier integrals to represent functions, which do not necessarily have a periodic feature.
1From the Fourier integral one can then derive the so-called complex Fourier transform or just Fourier
transform for short.
ˆ
f(ω) = 1
√2πZ∞
−∞
f(x)e−iωxdx. f (x) = 1
√2πZ∞
−∞
ˆ
f(ω)eiωxdω.
In the last set of notes we mention that these equations have a striking similarity to complex Fourier series
and that statements of energy and symmetry have analogies for Fourier transform. In the following we use
our knowledge of Fourier methods to gather insight into physical process that rely on Fourier analysis.
Let’s first begin with the following transform pair,
F{δ(t)}=1
√2πZ∞
−∞
δ(t)e−iωtdt =1
√2π⇐⇒ F−11
√2πff=δ(t).
This statement says if we wish to send a instantaneous pulse of information then its representation in the
frequency domain is a constant function. We take this to mean the following,
(1) A completely localized function requires an equal amount/amplitude of every possible frequency
of oscillation. Since the sum of the squares of these amplitudes is proportional to the energy of the
time-signal we conclude that this transmission would require an infinite amount of energy!
This transform highlights a fundamental property of Fourier transforms. That is, if a function is localized
in one domain then it is de-localized in the transformed domain. This relationship is the basis for the
Heisenberg uncertainty principle of quantum mechanics but also has a place any time the Fourier transform
concept is used. The two most important relations are:
•Position-momentum: In physics position and momentum are related by Fourier transform ∆x∆p≥
α∈R+and consequently if the position of a quantum particle is highly localized then its momentum
is de-localized. Thus if we know exactly where a particle is then we have no idea about where the
particle is going.
•Energy-time: In physics and engineering energy and time are related by Fourier transform ∆E∆t≥
α∈R+and consequently if the event takes place in an infinitesimal amount of time then it requires
an infinite amount of energy.
Though these statements eccentrically highlight important concepts they are motivated with non-rigorous
mathematical tricks involving the ‘delta-function.’ Maybe a more sensible function is given by,
f(t) = (A, −a < t < a,
0,otherwise,
1Again, we mention that periodicity can be reconstructed via delta ‘functions’. Thus making the Fourier integral
the more general representation.
1
