S No. Relaxation function $\begin{array}{l}i\left(t\right),\text{\hspace{0.17em}}t>0\\ t\equiv {x}_{0}+iy\end{array}$ Rate distribution function ${H}_{\lambda }\left(\lambda \right),\lambda >0$ ${H}_{\lambda }\left(\lambda \right)={\mathcal{L}}^{-1}\left\{i\left(t\right)\right\}$ 1 $i\left(t\right)=A$ : Constant function ${H}_{\lambda }\left(\lambda \right)=A\left(\delta \left(\lambda \right)\right)$ 2 $i\left(t\right)={t}^{-1}$ ${H}_{\lambda }\left(\lambda \right)=1,\lambda >0$ 3 $i\left(t\right)={t}^{-n}$ ${H}_{\lambda }\left(\lambda \right)=\frac{1}{\left(n-1\right)!}{\lambda }^{n-1}$ 4 $i\left(t\right)={\left(t+a\right)}^{-1}$ ${H}_{\lambda }\left(\lambda \right)={\text{e}}^{-a\lambda }$ 5 $i\left(t\right)={\left(t+a\right)}^{-n}$ ${H}_{\lambda }\left(\lambda \right)=\frac{1}{\left(n-1\right)!}{\lambda }^{n-1}{\text{e}}^{-a\lambda }$ 6 $i\left(t\right)=a{t}^{-1}{\left(t+a\right)}^{-1}$ ${H}_{\lambda }\left(\lambda \right)=1-{\text{e}}^{-a\lambda }$ 7 $i\left(t\right)={\left(t+a\right)}^{-1}{\left(t+b\right)}^{-1}$ ${H}_{\lambda }\left(\lambda \right)=\frac{1}{b-a}\left({\text{e}}^{-a\lambda }-{\text{e}}^{-b\lambda }\right)$ 8 $i\left(t\right)={\text{e}}^{-{\lambda }_{0}t}$ ${H}_{\lambda }\left(\lambda \right)=\delta \left(\lambda -{\lambda }_{0}\right)$ 9 $i\left(t\right)=t{\left({t}^{2}+a\right)}^{-1}$ ${H}_{\lambda }\left(\lambda \right)=\mathrm{cos}\left(\lambda \sqrt{a}\right)$ 10 $\sqrt{a}{\left({t}^{2}+a\right)}^{-1}$ ${H}_{\lambda }\left(\lambda \right)=\mathrm{sin}\left(\lambda \sqrt{a}\right)$ 11 $i\left(t\right)=\sqrt{a}{\left({\left(t+b\right)}^{2}+a\right)}^{-1}$ ${H}_{\lambda }\left(\lambda \right)={\text{e}}^{-b\lambda }\mathrm{sin}\left(\lambda \sqrt{a}\right)$ 12 $i\left(t\right)={\text{e}}^{-{\left(t/{\tau }_{0}\right)}^{\beta }}$ $\begin{array}{l}{H}_{\lambda }\left(\lambda \right)=\frac{{\tau }_{0}}{\text{π}}\underset{0}{\overset{\infty }{\int }}\left({\text{e}}^{\left(-{u}^{\beta }\mathrm{cos}\left(\beta \text{π}/2\right)\right)}\mathrm{cos}\left(\lambda {\tau }_{0}u-{u}^{\beta }\mathrm{sin}\left(\frac{\beta \text{π}}{2}\right)\right)\right)du\\ u=y/{\tau }_{0}\end{array}$ 13 $i\left(t\right)={\left(1+\left(1-\beta \right)\left(\frac{t}{{\tau }_{0}}\right)\right)}^{-1/\left(1-\beta \right)}$ $\begin{array}{l}{H}_{\lambda }\left(\lambda \right)=\frac{{\tau }_{0}}{\text{π}\left(1-\beta \right)}\underset{0}{\overset{\infty }{\int }}\left(du\right)\left(1+{u}^{2}{\right)}^{-1/\left(2\left(1-\beta \right)\right)}\mathrm{cos}\left(\frac{\lambda {\tau }_{0}u-{\mathrm{tan}}^{-1}u}{1-\beta }\right)\\ u=\left(1-\beta \right)y/{\tau }_{0}\end{array}$ 14 $i\left(t\right)={\left(1+{\left(\frac{t}{{\tau }_{0}}\right)}^{\alpha }\right)}^{-1};0<\alpha <1$ $\begin{array}{l}{H}_{\lambda }\left(\lambda \right)=\frac{{\tau }_{0}}{\pi }{\int }_{0}^{\infty }du\left(\frac{\left({u}^{\alpha }\mathrm{cos}\left(\frac{\alpha \pi }{2}\right)+1\right)\mathrm{cos}{\tau }_{0}u\lambda +\left({u}^{\alpha }\mathrm{sin}\left(\frac{\alpha \pi }{2}\right)\right)\mathrm{sin}{\tau }_{0}u\lambda }{{u}^{2\alpha }+2{u}^{\alpha }\mathrm{cos}\left(\frac{\alpha \pi }{2}\right)+1}\right)\\ u=y/{\tau }_{0}\end{array}$ 15 $\begin{array}{l}i\left(t\right)={E}_{\alpha }\left(-\xi \right),\xi =t/\tau \\ {E}_{\alpha }\left(-\xi \right)={\sum }_{k=0}^{\infty }\frac{{\left(-1\right)}^{k}}{\Gamma \left(\alpha k+1\right)}{\xi }^{k}\end{array}$ ${H}_{\lambda }\left(\lambda \right)=\frac{2}{\text{π}}{\int }_{0}^{\infty }{E}_{2\alpha }\left(-{y}^{2}\right)\mathrm{cos}\left(\lambda y\right)dy$ 16 $i\left(t\right)={E}_{1/2}\left(t/{\tau }_{0}\right)$ ${H}_{\lambda }\left(\lambda \right)=\frac{1}{\sqrt{\text{π}}}{\text{e}}^{-\left({\lambda }^{2}/4\right)}$