E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 11.7, pgs. 506-513
Lecture: Fourier Integral Module: 11
Suggested Problem Set: {3, 7, 9, 14, 15}March 30, 2009
E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 11.8, pgs. 513-517
Lecture: Fourier Sine/Cosine Transform Module: 11
Suggested Problem Set: {1, 5, 6}March 30, 2009
E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 11.9, pgs. 518-528
Lecture: Complex Fourier Transform Module: 11
Suggested Problem Set: {2, 3, 9, 14(a)}March 30, 2009
Quote of Lecture 11
Everybody’s talking ’bout the stormy weather and what’s a man do to but work out
whether it’s true? Looking for a man with a focus and a temper who can open up a map
and see between one and two. Time to get it before you let it get to you. Here he comes
now; stick to your guns and let him through.
Sonic Youth : Teenage Riot (1988)
1. Overview
Now that we have ended the study of Fourier series, which are used to represent piecewise continuous
functions that have the additional feature of being periodic, we as the question:
•Can these Fourier methods be applied to functions that do not have the extra structure of period-
icity?
Why would we like to do this? Well, we have seen already that Fourier series allow us to gather from a
function its particular frequencies of oscillation and also their associated amplitudes. With this description
one can then gain insight into the overall ‘energy’ of the function as well as how each of the oscillatory modes
contributes to this energy. Without the Fourier representation much of this information is inaccessible. If
we can port these methods over to functions without a periodic substructure then we may find similar inter-
pretations and consequently Fourier metho ds would be invaluable to an extremely large class of physically
relevant functions.
So, how should we go about this? Well, we must first notice that one of the fundamental assumption on
functions with a Fourier series representation is that they can b e defined by using a finite portion of the
real-line. This finite portion is considered to be the principle-period and from this information the rest of
the function is constructed by repeating the functions graph on this domain to the rest of R. If we consider
this principle-period to be the (−L, L), which is a finite portion of R, then we can destroy these concepts
by taking the limit L→ ∞. Letting Lbecome unbounded gives rise to a plausible heuristic derivation,
which results in the well-celebrated complex Fourier transform, or just Fourier transform for short. Though
the text chooses to consider first the limit, L→ ∞, to arrive at the Fourier integral and symmetrically
exploited to define the Fourier cosine/sine transforms, which upon reconsideration is used to define the more
general complex Fourier transform, we instead use the Fourier integral to go directly to the complex Fourier
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