Using Integrate and or FourierTransform

I have some general questions about Integrate and FourierTransform. Firstly with FourierTransform I am struggling to get anything meaningful from the result. For example: w0 = 24.; T = 0.2; F = FourierTransform[Sin[w0 t] Exp[-t/T],t,w] I get a result, but if I try and plot the absolute value of the result so Plot[Abs[F],{w,0,100}] I get nothing. In fact the Abs[F] just encapsulates the result of the FourierTransform.

So firstly, what am I doing wrong with FourierTransform?

If I however, do the Fourier transform manually with Integrate I have more success, so F = Integrate[Sin[2. Pi nu0 ] Exp[-t/T] Exp[-I 2 Pi nu t],{t,0,Infinity}], I can easily take the absolute value and plot it -- getting the result I would expect!

This would be fine, but I want to play with my model, adding some complexity, for example Integrate[Sin[2. Pi nu0(1 - Exp[-t/T]) ] Exp[-t/T] Exp[-I 2 Pi nu t],{t,0,Infinity}] so now my resonance frequency itself is a function of time. However Mathematica returns a convergence error i.e. the new function does not converge on $$0$$ or $$\infty$$

Can anyone shed any light on my two problems?

Just for clarity, I have also constrained my integrals with assumptions such that Assumptions-> w > 0 && t >0.

Finally, using Integrate can take a huge amount of time for example

Integrate[
Sin[2 \[Pi] \[Nu]0(1. - Exp[-t/\[Tau]]) t] Exp[-t/\[Tau]] Exp[- 2. I \[Pi] \[Nu] t ],
{t,0, +Infinity},
Assumptions->\[Nu] > 0 && t > 0
];
• Are you trying to plot a delta function? – chuy Apr 11 at 14:32
• @chuy this won't be a delta function, or it shouldn't be. – Q.P. Apr 11 at 14:37
• Ah I see, I think you actually want the FourierSinTransform. – chuy Apr 11 at 14:54
• @chuy Could you explain the difference? – Q.P. Apr 11 at 14:57
• @chuy FourierSinTransform is only applicable if I want $\omega$ as a variable. In my case it is not it is a function or a constant as it is a resonance frequency. – Q.P. Apr 11 at 15:07