= Scale-free density correlation transform
{title2=$\gamma=n+3,\quad -3<n<-1$}
For a statistically isotropic three-dimensional <matter power spectrum> $\mathcal P(k)=Ak^n$, the <Fourier transform> gives $\xi(r)=A(2\pi^2)^{-1}r^{-n-3}\int_0^\infty u^{n+1}\sin u\,du$. The <improper integral> converges at zero precisely when $n>-3$ and at infinity when $n<-1$; it is absolutely convergent only when $n<-2$. Within this range its positive coefficient is $\Gamma(n+2)\sin[\pi(n+2)/2]$, interpreted continuously as $\pi/2$ at $n=-2$. Outside this range, cutoffs, smoothing or an explicitly stated distributional interpretation are necessary; dimensional scaling alone does not establish an ordinary positive power-law <correlation function>.
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