Finding the fourier transform of...

Question

Find the Fourier transform of: https://i.imgur.com/h0Dh33n.png

Answer 1

f(x) as defined is a non-periodic function. Effectively it’s f(x)=1 for x∈(-5,5) and zero otherwise. So the following applies:

F(k)=(1/2π)∫f(x)e^(-ikx)dx[-∞,∞]=

(1/2π)∫e^(-ikx)dx[-5,5] because the function only exists in the narrow range [-5,5].

F(k)=(-1/(2πik))(e^(-5ik)-e^(5ik)).

e^(-5ik)=cos(5k)-isin(5k),

e^(5ik)=cos(5k)+isin(5k).

e^(-5ik)-e^(5ik)=-2isin(5k).

So F(k)=(1/(πk))sin(5k) (Fourier Transform).