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Definition of Fourier Transforms
If f(t) is a function of the real variable t, then the Fourier transform F(w) of f is given by
F(w) =  -infinity+infinity e -j w t f(t) dt.
In what follows, u(t) is the unit step function defined by
u(t) = 1 for t >= 0 and u(t) = 0 for t < 0. (see figure below).
i = sqrt (-1), the imaginary unit.
Table of Fourier Transforms.
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f(t)
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F(w)
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u(t) e -a t , a > 0
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1 / (a + i w)
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1 for - a <= t <= a
and 0 otherwise
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2 sin (w a) / w
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A (constant)
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2 p A d (w)
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d (t)
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1
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d (t - a)
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e -i w a
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cos (a t)
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p [ d (w + a) + d (w - a) ]
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sin (a t)
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(p / i) [ d (w - a) - d (w + a) ]
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e i a t
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2 p [ d (w - a) ]
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f'(t)
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i w f(w)
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f"(t)
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(i w) 2 f(w)
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t f(t)
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i d [f(w)] / dw
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t 2 f(t)
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i 2 d 2[f(w)] / dw 2
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More references on
integrals and their applications in calculus.
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