\begin{align*} \frac{1}{z^{\alpha}} & =\frac{1}{\Gamma\left(\alpha\right)}\frac{\Gamma\left(\alpha\right)}{z^{\alpha}}\\ & =\frac{1}{\Gamma\left(\alpha\right)}\mathcal{L}_{t}\left[H\left(t\right)t^{\alpha-1}\right]\left(z\right)\\ & =\frac{1}{\Gamma\left(\alpha\right)}\int_{-\infty}^{\infty}H\left(t\right)t^{\alpha-1}e^{-zt}dt\\ & =\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{\infty}t^{\alpha-1}e^{-zt}dt \end{align*}