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Fourier Transform of exponential and heaviside function

  • Thread starter katiandss
  • Start date
1. The problem statement, all variables and given/known data
Compute the Fourier transform of

[itex]\phi(t)[/itex]=(e^(-at))H(t)

where H(t) is the Heaviside step function


2. Relevant equations



3. 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

Gold Member
1,198
12
The Fourier Transform is defined on a function f(t) as:

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

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.
 

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