A few days ago I came across a paper (don't remember the name) that claimed that the auto-correlation function of a process defined by a "by-part" power-law (like here : PSD), which can be described by several exponents, also follows a power-law.
Now, the autoccorelation function is defined as the inverse fourier transform of the power-spectral density, so I wanted to see if this claim is true by calculating it myself. We know, for instance, that this is true for a pure power law since the fourier transform : $$\hat f(t)=\int_0^\infty t^\alpha e^{i\omega t}d\omega.$$ is $$(-it)^{-(\alpha+1)}\Gamma(\alpha+1).$$ So it might be the case for a "by-part" example. I tried computing the fourier transform myself, of the following : $$\hat f(t)=\int_0^{w_c} t^{\alpha_1} e^{-i\omega t}d\omega + \int_{w_c}^\infty t^{\alpha_2} e^{-i\omega t}d\omega.$$ With ${w_c}$ being the point where the slope exponent changes, $\alpha_1$ and $\alpha_2$ being the two exponents. I tried changing the variables to get close to a gamme function without succes and I also tried solving them with help of the residue theorem without better success...
Does anyone has an idea how I can solve this integral ?