Fourier Transform of exponential and heaviside function

[katiandss]

Homework Statement

Compute the Fourier transform of

\phi(t)=(e^(-at))H(t)

where H(t) is the Heaviside step function

Homework Equations

The Attempt at a Solution

I am stuck in an attempt at the solution, I am confused at how the heaviside step function factors in and think that it may just affect the upper and lower limits of the integral, but am not sure. I am looking for direction on how to approach the problem or at least set it up.

 

[Char. Limit]

The Fourier Transform is defined on a function f(t) as:

\int_{-\infty}^\infty f(t) e^{2 \pi i t \omega} dt

Now, try plugging in f(t)=e^(-at)H(t) into this definition. Remember that H(t) is defined to be 0 for all negative t and 1 for all positive t, so try splitting the integral into two integrals: one with lower bound -infinity and upper bound 0, and the other with lower bound 0 and the upper bound infinity. Then apply the definition of H(t) and it should become easy.