# rules for Laplace transform

If  $\mathcal{L}\{f(t)\}=F(s)$,  then

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

• $\mathcal{L}\{f(\frac{t}{a})\}\;=\;a\,F(as)$   for  $a>0$.

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

 $\mathcal{L}\{e^{at}f(t)\}=\int_{0}^{\infty}\!e^{-st}e^{at}f(t)\,dt=\int_{0}^{% \infty}\!e^{-(s-a)t}f(t)\,dt=\int_{0}^{\infty}\!e^{-rt}f(t)\,dt=F(r)=F(s\!-\!a).$

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

 $\mathcal{L}\{f(\frac{t}{a})\}=\int_{0}^{\infty}\!e^{-st}f(\frac{t}{a})\,dt=a\!% \int_{0}^{\infty}\!e^{-sau}f(u)\,du=a\!\int_{0}^{\infty}\!e^{-ru}f(u)\,du=aF(r% )=a\,F(as).$
Title rules for Laplace transform RulesForLaplaceTransform 2013-03-22 18:31:08 2013-03-22 18:31:08 pahio (2872) pahio (2872) 7 pahio (2872) Derivation msc 44A10 TableOfLaplaceTransforms