The function

The Differential Transform

Where

1

$f_{\left(t\right)}$

$d{f}_{\left(t\right)}=\frac{1}{dt}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{\frac{-\tau }{dt}}{f}_{\left(\tau \right)}d\tau }$

2

$h\left(t\right)=f\left(t\right)\ast y\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}{f}_{\left(t-\tau \right)}{y}_{\left(\tau \right)}d\tau }$

$d{h}_{\left(t\right)}=d{f}_{\left(t\right)}dtd{y}_{\left(t\right)}$

3

$f_{\left(t-\tau \right)}$ , $t\ge \tau$

$e^{-\frac{\tau }{dt}}d{f}_{\left(t\right)}$

4

$e^{-at}f\left(t\right)$

$\frac{1}{1+dat}G\left(\frac{dt}{1+dat}\right)$ , $G=df$

5

unit impulse ${\delta }_{\left(t\right)}$

$1/dt$

6

$t^n/n!$

$d_t^n$

7

Constant: C

C

8

$e^{-\alpha t}$

$1/\left(1+d\alpha t\right)$

9

$\sin \left(\omega t\right)$

$d\omega t/\left(1+d_{\omega t}^2\right)$

10

$\cos \left(\omega t\right)$

$1/\left(1+d_{\omega t}^2\right)$

11

$\sinh \left(\omega t\right)$

$d\omega t/\left(1-d_{\omega t}^2\right)$

12

$t\sinh \omega t$

$\frac{2d_{\omega t}^2}{\omega {\left(1-d_{\omega t}^2\right)}^2}$

13

$\cosh \left(\omega t\right)$

$1/\left(1-d_{\omega t}^2\right)$

14

$t\cosh \omega t$

$dt\frac{1+d_{\omega t}^2}{{\left(1-d_{\omega t}^2\right)}^2}$

15

$t^n{e}^{-\alpha t}$

$d_t^n/{\left(1+\alpha dt\right)}^{n+1}$

16

$t\sin \omega t$

$\frac{2d_{\omega t}^2}{\omega {\left(1+d_{\omega t}^2\right)}^2}$

17

$\sin \omega t/t$

$\arctan \left(d\omega t/dt\right)$

18

$t\cos \omega t$

$dt\frac{1-d_{\omega t}^2}{{\left(1+d_{\omega t}^2\right)}^2}$

19

$\frac{1-\cos \omega t}{t}$

$\frac{\ln \sqrt{1+d_{\omega t}^2}}{dt}$

20

$e^{-\alpha t}\sin \omega t$

$\frac{\omega dt}{d_{{\omega }_{n}t}^2+2\alpha dt+1}$

$\alpha +i\omega ={\omega }_n{e}^{i\theta }$

21

$e^{-\alpha t}\cos \omega t$

$\frac{1+\alpha dt}{d_{{\omega }_{n}t}^2+2\alpha dt+1}$

${\omega }_n=\sqrt{{\alpha }^2+{\omega }^2}$

22

$\frac{e^{-\alpha t}}{\sin \theta }\sin \left(\theta -\omega t\right)$

$\frac{1}{d_{{\omega }_{n}t}^2+2\alpha dt+1}$

$\theta =\arctan \left(\omega /\alpha \right)$

23

$1-\frac{e^{-\alpha t}}{\sin \theta }\sin \left(\omega t+\theta \right)$

$\frac{d_{{\omega }_{n}t}^2}{d_{{\omega }_{n}t}^2+2\alpha d{\omega }_{n}t+1}$

$\alpha ={\omega }_n\cos \theta$

24

${\omega }_n{e}^{-\alpha t}\cos \left(\omega t-\theta \right)$

$\frac{{\omega }_n^{2}dt+\alpha }{d_{{\omega }_{n}t}^2+2\alpha dt+1}$

$\omega ={\omega }_n\sin \left(\theta \right)$