How to let Mathematica return impulse or Dirac delta functions when computing integrals?

Question

For example, let's say I want to compute the (continuous-time) Fourier transform of the signal/function $\cos{(3t)}$, which is given by the following improper integral:

$\displaystyle\int_{-\infty}^{\infty} \cos{(3t)} e^{-j\omega t} \, \mathrm{dt} \tag 1$

whose value is:

$\pi \, \delta{(\omega - 3)} + \pi \, \delta{(\omega + 3)} \tag 2$

The corresponding Mathematica code to compute the integral of (1) would be Integrate[Cos[3*t]*Exp[-I*w*t], {t, -Infinity, Infinity}, Assumptions -> {w \[Element] Reals}]. However, after executing such code Mathematica returns the message "Integrate: Integral of $e^{-i t w} \text{Cos}{[3 t]}$ does not converge on {$-\infty$,$\infty$}." How can I tell Mathematica to use the impulse/Dirac delta function during the computation of the integral, in order to return (2) or something similar?

Answer 1

If you use Dirac or Heaviside functions explicitly in your expression, Mathematica figures out that you're working with generalized functions. Unfortunately, it doesn't always work the other way: Mathematica won't volunteer to use generalized functions unless you're explicitly using Fourier/Laplace transforms. So, if you're working with Fourier integrals, and want results in terms of generalized functions, you need to use FourierTransform rather than Integrate.