The Fourier transform of exp(-x)*heaviside(x)

Question

I'm trying to understand the Fourier transform of Y=exp^-x. Since the term tends to -infinity I have to multiply Y by the heaviside function to set everything below 0 to 0 so I can successfully perform the Fourier transform. In matlabs, Im using ezplot to only plot the absolute values and not the imaginary components. my code is as follows

f = exp(-x)*heaviside(x); 
X=fourier(f, x, w);
ezplot(abs(X))
grid on
hold on
ezplot(f, [-2, 2]);
hold off

The plot I'm getting is

Is this because my imaginary components are being set to zero and the real parts are not. Since I am only plotting the absolute values this would make sense. Can anyone shed some light on this?

Answer 1

$$ \int_0^{\infty} e^{-x} e^{-ikx} \, dx = \left[\frac{-e^{-x}e^{ikx}}{1+ik} \right]_0^{\infty} = \frac{1}{1+ik} $$ The absolute value of this is $$ \sqrt{\frac{1}{1+ik}\frac{1}{1-ik^*}} = \frac{1}{\sqrt{1+|k|^2}} $$ which produces the graph you have.