$f\left(t\right)\ to\ F\left(s\right)$

$F\left(s\right)\ to\ f\left(t\right)$

$f\left(t\right)\ to\ f'\left(t\right)$

$f"\left(t\right)\ to\ f'\left(t\right)$

 $\frac{1}{s^3}$  

 $\frac{2}{s^3}$  

 $\frac{6}{s^4}$  

 $\frac{3}{s^3}$  

$F\left(s\right)=\int_0^{\infty}e^{-st}f\left(t\right)dt$

$F\left(s\right)=\int_0^tf\left(t\right)g\left(t-u\right)$

Linearity Property

First Shifting Property

Convolution Theorem

Second Shifting Property

 $a=2,\ f\left(t\right)=e^{2t}$  

 $a=4,\ f\left(t\right)=\cos\ t$  

 $a=2,\ f\left(t\right)=\sin\ t$