Does Wolframalpha have commands that (easily) allow me to supply a mathematical function and, then, have outputted a plot of the Fourier spectrum? Or do I have to go through the process of myself of Fourier transforming and extracting amplitudes from that and, then, plotting?
What have you tried? This is the sort of thing one should first experiment with, and go to a forum if nothing suitable was found.
Hi Daniel, I've been using the Fourier transform command to obtain a frequency domain expression of my function, then taken its absolute value by multiplying by the complex conjugate, then plotted the result. It's all just standard stuff, so I guess I thought there would be a sort of signal-processing toolbox to do this, and other related things. I'm just exploring.
By the way, why is that the following command gives me a result (FT of a Dawson function):
Fourier transform of Dawson(sqrt(t)*theta(t))
but the following (same thing, but with time multiplied by a constant:
Fourier transform of Dawson(sqrt(at)theta(t))
does not? This difference is just an example of how sometimes Wolframalpha works well, but at other times, for reasons I don't understand, it doesn't.
Could just request the absolute value. This worked fine for me:
WolframAlpha["abs of fourier transform Dawson(sqrt(t)theta(t))"]
It even showed a plot.
The second issue probably comes down to having a parameter that W|A balks on. Using specific numbers seems to work fine. The Wolfram Language can handle the general parameter case using the Assumptions option.
FourierTransform[DawsonF[Sqrt[at]]HeavisideTheta[t], t, w,
Assumptions -> a > 0]
(* Out= (Sqrt[a] (I w + Abs[w]))/(4 (a - I w) Abs[w]^(3/2)) *)
General remark: Posting specific examples makes it easier for others to do diagnoses and provide relevant responses. This post had such an example and that of course is quite helpful.