I've been using Wolfram Alpha for the Fourier Transform because our professor doesn't require us to use the actual definition to solve for the Fourier Transform. Rather, he doesn't mind if we use textbook tables of solved Fourier Transforms and their properties.

We were assigned a few problems that asked us for the Fourier Transforms of particular signals.

Say, the Fourier Transform of $1$ in terms of $\omega$ is $2\pi$. However, when I double checked to make sure it was correct in Wolfram Alpha, the result was $\sqrt{2\pi}$... Why? There appears to be an inconsistency between the textbook and Wolfram Alpha. Another example is the Fourier Transform of $\cos(\omega_o t)$... In the textbook it says, in terms of $\omega$, $\pi[\delta(\omega-\omega_o)+\delta(\omega+\omega_o)]$ but again in Wolfram Alpha, it says, $\sqrt{\frac{\pi}{2}}[\delta(\omega-\omega_o)+\delta(\omega+\omega_o)]$.

Does Wolfram Alpha utilize a different definition or algorithm of the Fourier Transform that I'm not aware of? I'm really confused. I verbatim type in on Wolfram Alpha "Fourier Transform of (some function)". I'm making a heavy assumption that it believes I want to include a Heaviside step function along with the main function I would like to take the Fourier Transform for?

  • 2
    $\begingroup$ There are different conventions for the FT differing in their normalization constants. WA uses what is called the unitary convention by default. $\endgroup$ – Ian Apr 1 '18 at 22:55
  • $\begingroup$ Note also the footnote in the input interpretation section. If you hover your cursor over it, you will reveal links to the definition and Wolfram Language documentation which more explicitly indicate the convention used by WA. $\endgroup$ – user170231 Apr 1 '18 at 23:57

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