S No.

Relaxation function

$\begin{array}{l}i\left(t\right),t>0\\ t\equiv x_0+iy\end{array}$

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

$i\left(t\right)=t{\left(t^2+a\right)}^{-1}$

$H_{\lambda }\left(\lambda \right)=\cos \left(\lambda \sqrt{a}\right)$

10

$\sqrt{a}{\left(t^2+a\right)}^{-1}$

$H_{\lambda }\left(\lambda \right)=\sin \left(\lambda \sqrt{a}\right)$

11

$i\left(t\right)=\sqrt{a}{\left({\left(t+b\right)}^2+a\right)}^{-1}$

$H_{\lambda }\left(\lambda \right)={\text{e}}^{-b\lambda }\sin \left(\lambda \sqrt{a}\right)$

12

$i\left(t\right)={\text{e}}^{-{\left(t/{\tau }_0\right)}^{\beta }}$

$\begin{array}{l}{H}_{\lambda }\left(\lambda \right)=\frac{{\tau }_0}{\text{π}}{\displaystyle \underset{0}{\overset{\infty }{\int }}\left({\text{e}}^{\left(-u^{\beta }\cos \left(\beta \text{π}/2\right)\right)}\cos \left(\lambda {\tau }_{0}u-u^{\beta }\sin \left(\frac{\beta \text{π}}{2}\right)\right)\right)du}\\ u=y/{\tau }_0\end{array}$

13

$i\left(t\right)={\left(1+\left(1-\beta \right)\left(\frac{t}{{\tau }_0}\right)\right)}^{-1/\left(1-\beta \right)}$

$\begin{array}{l}{H}_{\lambda }\left(\lambda \right)=\frac{{\tau }_0}{\text{π}\left(1-\beta \right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}\left(du\right)(1+u^2}{)}^{-1/\left(2\left(1-\beta \right)\right)}\cos \left(\frac{\lambda {\tau }_{0}u-{\tan }^{-1}u}{1-\beta }\right)\\ u=\left(1-\beta \right)y/{\tau }_0\end{array}$

14

$i\left(t\right)={\left(1+{\left(\frac{t}{{\tau }_0}\right)}^{\alpha }\right)}^{-1};0<\alpha <1$

$\begin{array}{l}{H}_{\lambda }(\lambda )=\frac{{\tau }_0}{\pi }{\displaystyle {\int }_0^{\infty }du\left(\frac{\left(u^{\alpha }\cos \left(\frac{\alpha \pi }{2}\right)+1\right)\cos {\tau }_{0}u\lambda +\left(u^{\alpha }\sin \left(\frac{\alpha \pi }{2}\right)\right)\sin {\tau }_{0}u\lambda }{u^{2\alpha }+2u^{\alpha }\cos \left(\frac{\alpha \pi }{2}\right)+1}\right)}\\ u=y/{\tau }_0\end{array}$

15

$\begin{array}{l}i\left(t\right)=E_{\alpha }\left(-\xi \right),\xi =t/\tau \\ E_{\alpha }\left(-\xi \right)={\displaystyle {\sum }_{k=0}^{\infty }\frac{{\left(-1\right)}^k}{\Gamma \left(\alpha k+1\right)}{\xi }^k}\end{array}$

$H_{\lambda }\left(\lambda \right)=\frac{2}{\text{π}}{\displaystyle {\int }_0^{\infty }{E}_{2\alpha }\left(-y^2\right)\cos \left(\lambda y\right)dy}$

16

$i\left(t\right)=E_{1/2}\left(t/{\tau }_0\right)$

$H_{\lambda }\left(\lambda \right)=\frac{1}{\sqrt{\text{π}}}{\text{e}}^{-\left({\lambda }^2/4\right)}$