# Show that if $L\{F(t)\} = f(s)$ then $L\{F(at)\} = \frac{1}{a} f(\frac{s}{a})$

I'm trying to answer this question and I just don't know how to finish it.

I've tried integrating the $te^{-st}$ by parts and then multiplying it by $\frac{1}{a}$ but it doesn't show the answer I should expect. Any help would be so greatly appreciated.

If $\hat{F} = {\cal L} F$, and $\phi(t) = F(a t)$ with $a>0$, we have ${\cal L} \phi (s) = \int_0^\infty F(a t) e^{-st} dt$. Using the substitution $u = a t$, we obtain ${\cal L} \phi (s) = \int_0^\infty F(u) e^{-s \frac{u}{a}} \frac{1}{a}du = \frac{1}{a} \int_0^\infty F(t) e^{-\frac{s}{a}t}dt = \frac{1}{a} {\cal L} \hat{F} (\frac{s}{a})$.