# E its properties are the same at every point x x and

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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.

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