I cannot find reference (paper) of this relation $u(t)t^{α} ↔^{FT} f^{-(α+1)}$
You will find a &quot;proof&quot; of principle in math-stackexchange: Fourier transform of power function $t^\alpha$ (the title lacks the mention of the unit step function $u(t)$). At least formally, the Fourier transform of one-sided power law turns into a Laplace transform, and looks a lot like the Gamma function (the one related to the factorial).
You can find a similar description on page 276 of the chapter  Appendix A. Mathematical Background, Power law.
What I find troubling in this &quot;simple proof&quot; is that the convergence and well-behaved properties of the power-law functions are not satisfied. They are not integrable in the traditional sense, so it should be treated with tempered distributions or generalized functions, and Schwartz functions decaying faster than powers.
You will find a clean demonstration in Terence Tao blog: 245C, Notes 3: Distributions. 3. Tempered distributions, around Equation 9. In dimension $d$ ($d=1$ for you), the formula reads:
$$\widehat{u(t)t^\alpha}(f) = \frac{\pi^{-(d-\alpha)/2}\Gamma((d-\alpha)/2)}{\pi^{-\alpha/2}\Gamma(\alpha/2)}|f|^{-(d-\alpha)}$$
[Nota: this is a follow-up of question: Can I simplify $x=\frac{\ln(|fft((ifft(\sqrt{(f^{-5/3})}))^{2})|)}{\ln(f)}$]