How can I find the Fourier transform of 1/(t+2)^2?
1 Answer
Jun 21, 2018
Explanation:
There are various forms, so specifically using this definition:
#bbbF_t[f(t)] (omega) = int _(-oo)^oo f(t) e^(-i omega t) dt = F(omega) #
With Fourier pairs written:
#f(t) harr F(omega)#
Start by evaluating
This is well-known Fourier pair, and easy to evaluate:
# sgn(t) harr -(2 i)/omega#
The significance is the duality property of the Fourier transform:
#f(t) harr F(omega) implies F(t) harr 2 pi f(-omega)#
So:
#bbbF [ -(2 i)/t] = 2 pi sgn(-omega) = - 2 pi sgn(omega)#
By linearity :
#bbbF [ 1/t] = 1/(2 i) 2 pi sgn(omega) = - i pi sgn(omega)#
From the derivative property:
By linearity :
We want:
#bbbF[f(t-a)] = e^(- i a omega) F(omega)#
With