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A Fourier Transform of a piecewise function

  1. Jul 24, 2016 #1
    Here is the Problem Statement : Find Fourier Transform of the piecewise function


    Can someone sheds some lights on how to start solving this?

  2. jcsd
  3. Jul 24, 2016 #2
    The Fourier transform of your function f(t) is given as:

    $$ F\left[f(t)\right] = \int_{-\infty}^{\infty} dt e^{i\omega t}f(t) = \int_{-\tau}^{0} -e^{i\omega t}dt + \int_{0}^{\tau} e^{i\omega t}dt $$

    In the last step, I made use of the fact that f(t) is 0 elsewhere. As a final step, one can perform a simple integration to solve for the Fourier transform of f(t).
  4. Jul 24, 2016 #3
    Thanks Absalonsen! Is it e^(iwt) or e^(-iwt)?Let me know.
  5. Jul 24, 2016 #4
    np. It is usually a convention to determine the sign of the exponential in Fourier transform. In physics, forward Fourier transform from time to frequency space is carried out by ##e^{-iwt}##, while forward Fourier transform from real space to momentum space contains ##e^{ikx}##.
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