fourier transforms with Woflram Alpha

Question

I'm studying Fourier transform at the moment and I've noticed some weird irregularities between my textbook and Wolfram Alpha's answers.

Say I ask Wolfram the following:

fourier transform delta(t-1)

My textbook claims the fourier transform of this is:

exp(-j*2pi*f)

but then Woflram says otherwise:

It's not only with this specific transform. I noticed it with several others. Why the difference?

Answer 1

That is because there are several conventions followed in defining the Fourier Transform. By default, Mathematica computes $$ \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty f(t)e^{-i\omega t}\mathrm{d}t $$ when you ask for the Fourier Transform. But your Textbook defines it as $$ \int\limits_{-\infty}^\infty f(t)e^{-i\omega t}\mathrm{d}t $$ For getting the Fourier Transform defined as above, Try
FourierTransform[DiracDelta[t-1],t,\[Omega], FourierParameters -> {1,-1}]

You can read more about the different conventions and their corresponding parameters in the documentation.