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Unformatted text preview: 1/T )eit/L +
f (2/L)e2it/L + . . . .
2πL
2πL
2πL
In the limit as L → ∞, it can be shown that this last sum approaches an
improper integral, and our formula becomes
+ f (t) = 1
2π ∞ ˆ
f (ξ )eiξt dξ. (3.23) −∞ Equation (3.23) is called the Fourier inversion formula . If we make use of
Euler’s formula, we can write the Fourier inversion formula in terms of sines
and cosines,
f (t) = 1
2π ∞ i
ˆ
f (ξ ) cos ξtdξ +
2π
−∞ ∞ ˆ
f (ξ ) sin ξtdξ, −∞ a superposition of sines and cosines of various frequencies.
Equations (3.22) and (3.22) allow one to pass back and forth between a given
function and its representation as a superposition of oscillations of various frequencies. Like the Laplace transform, the Fourier transform is often an eﬀective
tool in ﬁnding explicit solutions to diﬀerential equations.
Exercise:
3.5.1. Find the Fourier transform of the function f (t) deﬁned by
f (t) = 1, if −1 ≤ t ≤ 1,
0, otherwise. 80 Chapter 4 Partial Diﬀerential
Equations
4.1 Overview A partial diﬀerential...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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