The function

The series

1

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

$A_n$

2

$\alpha e^t$

$\alpha$

3

$t{e}^t$

n

4

$\alpha e^{\beta t}$

$\alpha {\beta }^n$

5

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

$n{\alpha }^n$

6

$\sin \left({\omega }_{0}t\right)$

${\omega }_0^n\sin \left(n\frac{\pi }{2}\right)$

7

$\cos \left({\omega }_{0}t\right)$

${\omega }_0^n\cos \left(n\frac{\pi }{2}\right)$

8

${\displaystyle \underset{0}{\overset{t}{\int }}{f}_{\left(t\right)}{d}_t}$

$A_{n-1}$ , $n-1\ge 0$

9

${\displaystyle \underset{0}{\overset{t}{\int }}\stackrel{m}{\cdots }{\displaystyle \underset{0}{\overset{t}{\int }}{f}_{\left(t\right)}}{d}_t}\stackrel{m}{\cdots }{d}_t$ , m times integration

$A_{n-m}$ , $n-m\ge 0$

10

$\frac{1}{t}{\displaystyle \underset{0}{\overset{t}{\int }}{f}_{\left(t\right)}{d}_t}$

$\frac{1}{n+1}{A}_n$

11

${\displaystyle \underset{0}{\overset{t}{\int }}\left(f_{\left(t\right)}/t\right)d_t}$

$\frac{1}{n}{A}_n$

12

$t^m\frac{d^m}{d_t^m}f\left(t\right)$ , m times derivation

$\frac{n!}{\left(n-m\right)!}{A}_n$