of $x$, such that $\Theta(0)$ is finite, hence it does not spoil the power-law growth at the origin. For large $x$, it may present different asymptotic behaviors depending on the value of $s$:

[**A.1**]{}: If $s\leq 0$, $\Theta(x)$ decays as pure [*exponential*]{} of $x$. Therefore, the asymptotic decay is finally dominated by this exponential factor.

[**A.2**]{}: If $0<s<1/2$, $\Theta(x)$ behaves asymptotically as a [*stretched exponential*]{} with argument $x^{1-2s}$. That is, the tail, although a power law for moderate $x$, becomes asymptotically dominated by a stretched exponential decay.

[**A.3**]{}: If $s>1/2$, $\Theta(x)$ tends to a positive value, therefore, in this instance, the tail remains [*power-law*]{}.

There, by switching $s$, one tunes the tail type, being a power-law for $s\ge 1/2$. In the threshold case $s=1/2$, we have $_2F_1(1,1,2,-z)\equiv\ln(1+z)/z$, then we get the explicit expression $$\label{A}
\rho_{\rm A}(x) = \rho_o  
\frac{x^{\gamma_\alpha\theta}}
{[1+\lambda^2x]^{\gamma_\alpha\theta+\gamma_\mu+1} }  \; .$$

Thus, the case $r=0$, $s\ge1/2$ allows one to model empirical PDFs with twofold power-law behavior.

[**B**]{}: In the case $r=-1$, the normalization condition requires: $s\ge 1/2$, or also, if $\gamma_\alpha>1$, $s\le 0$ is allowed. The PDF has the form $$\label{IIIm}
\rho(x) = \rho_o  
\frac{x^{-\gamma_\alpha} \Theta(x)}
{[1+\lambda^2x^{2s}]^{-\gamma_\alpha/(2s)+1} }  \; ,$$ where, $\Theta(x)\equiv\exp[-\gamma_\alpha\theta\,_2F_1({\scriptstyle -\frac{1}{2s},
1,-\frac{1}{2s}+1},-\lambda^2x^{2s})/x]$ tends to a finite 