I need more help understanding Fourier Transforms. I know that they transform a function from the time domain to the frequency domain and vice versa, but the short cuts to solve them just straight up confuse me.

http://www.cse.unr.edu/~bebis/CS474/.../FT_Pairs1.pdf

This list of relations makes sense, but it's so hard for me to apply this to actual functions for some reason... I'd almost rather just do the integrals.

For example, every wave is of the form (except for the first exponential, which can always be different)

f(t) = exp[-(t^2)/2]exp(i*w*t)

The exp(i*w*t) part is the plane wave, correct? What is exp[-(t^2)/2] called? This term localizes the wave to keep it from being a plane wave, but I don't know what to call it.

Do you take the fourier transform of the whole f(t), or just the localizing term?

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 Quote by cytochrome For example, every wave is of the form (except for the first exponential, which can always be different) f(t) = exp[-(t^2)/2]exp(i*w*t) The exp(i*w*t) part is the plane wave, correct? What is exp[-(t^2)/2] called? This term localizes the wave to keep it from being a plane wave, but I don't know what to call it.
Where are you getting this part from? I suspect you are misunderstanding something with this. Fourier transforms allow us to decompose a function into contributions of individual waves of the form

$$F(\omega)e^{i\omega t}$$

$F(\omega)$ can be any function (well, practically any). A given function $f(t)$ determines $F(\omega)$ and vice versa.

If you want to truly understand Fourier transforms, I recommend getting a good grasp of Fourier series first.