Table of Fourier Transforms
Definition of Fourier Transforms
If f(t) is a function of the real variable t, then the Fourier transform F(ω) of f is given by the integral
F(ω) = ∫_{∞}^{+∞} e^{  j ω t} f(t) dt
where j = √(1), the imaginary unit.
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).
Table of Fourier Transforms.
f(t)

F(ω)

u(t) e^{ a t} , a > 0

1 / (a + j ω)

1 for  a ≤ t ≤ a
and 0 otherwise

2 sin (ω a) / ω

A (constant)

2 π A δ (ω)

δ (t)

1

δ (t  a)

e^{  j ω a}

cos (a t)

π [ δ(ω + a) + δ(ω  a) ]

sin (a t)

 j π [ δ (ω  a)  δ (ω + a) ]

e ^{ j a t}

2 π [δ(ω  a) ]

f '(t)

j ω F (ω)

f ''(t)

(j ω)^{ 2} F(ω)

t f(t)

j d [F(ω)] / dω

t^{ 2} f(t)

j^{ 2} d^{ 2}[F(ω)] / dω^{ 2}

More references on
integrals and their applications in calculus.
