rules for Laplace transform

If  $ℒ\{f(t)\}=F(s)$,  then

• $ℒ\{e^{at}f(t)\}=F(s-a)$  for  $s>a$,

• $ℒ\{f(\frac{t}{a})\}=aF(as)$   for  $a>0$.
  

For deriving these rules, we start from the definition of Laplace transform.  In the first case, we shall use the notation  $s-a=r$:

$$ℒ\{e^{at}f(t)\}={\int }_0^{\mathrm{\infty }}{e}^{-st}{e}^{at}f(t)𝑑t={\int }_0^{\mathrm{\infty }}{e}^{-(s-a)t}f(t)𝑑t={\int }_0^{\mathrm{\infty }}{e}^{-rt}f(t)𝑑t=F(r)=F(s-a).$$

In the second case, we make the change of variable  $\frac{t}{a}=u$  and later use the notation  $sa=r$:

$$ℒ\{f(\frac{t}{a})\}={\int }_0^{\mathrm{\infty }}{e}^{-st}f(\frac{t}{a})𝑑t=a{\int }_0^{\mathrm{\infty }}{e}^{-sau}f(u)𝑑u=a{\int }_0^{\mathrm{\infty }}{e}^{-ru}f(u)𝑑u=aF(r)=aF(as).$$