The Fourier transform of f .t / is given by F .!/ D Z 1 ?1 f

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The Fourier transform of f .t / is given by F .!/ D Z 1 ?1 f .t /e?i!t dt; which, for suitable functions f; can be regar

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The Fourier transform of f .t / is given by F .!/ D Z 1 ?1 f .t /e?i!t dt; which, for suitable functions f; can be regarded as the extension of the Laplace transform to the interval .?1;1/ with s replaced by i!. The inverse Fourier transform of F .!/ is then given by f .t / D 1 2 Z 1 ?1 F .!/ei!t d!: (a) Use these two integrals to express f .t / as an iterated double integral. Assuming that the order of the integrals can be reversed, find an integral representation of the Dirac delta function ?.t /. (b) Assuming that the usual properties of definite integrals hold when representing a generalized function, show that ?.t / acts like any even function, that is, ?.?t / D ?.t /.

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