Table of Fourier Transforms

Table of Fourier Transforms

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) = -∞+∞ 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).

unit step function


i = sqrt (-1), the imaginary unit.

Table of Fourier Transforms.

f(t) F(w)
u(t) e -a t , a > 0 1 / (a + i w)
1 for - a <= t <= a
and 0 otherwise
2 sin (w a) / w
A (constant) 2 p A d (w)
d (t) 1
d (t - a) e -i w a
cos (a t) p [ d (w + a) + d (w - a) ]
sin (a t) (p / i) [ d (w - a) - d (w + a) ]
e i a t 2 p [ d (w - a) ]
f'(t) i w f(w)
f"(t) (i w) 2 f(w)
t f(t) i d [f(w)] / dw
t 2 f(t) i 2 d 2[f(w)] / dw 2


More references on
integrals and their applications in calculus.

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