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Unformatted text preview: , extended to be periodic of period 2π .
b. Use (3.19) to ﬁnd the real form of the Fourier series.
3.4.4. Show that if a function f : R → R is smooth and periodic of period 2πL,
we can write the Fourier expansion of f as
f (t) = . . . + c−2 e−2it/L + c−1 e−it/L + c0 + c1 eit/L + c2 e2it/L + . . . ,
where
ck = 3.5 1
2πL πL f (t)e−ikt/L dt. −πL Fourier transforms* One of the problems with the theory of Fourier series presented in the previous
sections is that it applies only to periodic functions. There are many times when
one would like to divide a function which is not periodic into a superposition of
sines and cosines. The Fourier transform is the tool often used for this purpose.
The idea behind the Fourier transform is to think of f (t) as vanishing outside
a very long interval [−πL, πL]. The function can be extended to a periodic
function f (t) such that f (t + 2πL) = f (t). According to the theory of Fourier
series in complex form (see Exercise 3.4.4),
f (t) = . . . + c−2 e−2it/L + c−1 e−it/L + c0 + c1 eit/L + c2 e2it/L + . . . ,
where the ck ’s are the complex numbers.
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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