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Unformatted text preview: 1/T )eit/L +
f (2/L)e2it/L + . . . .
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
f (t) = 1
2π ∞ i
f (ξ ) cos ξtdξ +
−∞ ∞ 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.
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
4.1 Overview A partial diﬀerential...
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This document was uploaded on 01/12/2014.
- Winter '14