- #1
Saladsamurai
- 3,020
- 7
Homework Statement
Show that if
[tex]\text{L}\[f(t)] = F(s) \text{ then } \text{L}\[f(at)] = \frac{1}{a}F(\frac{s}{a})[/tex]
Homework Equations
Definition of Laplace
The Attempt at a Solution
By definition,
[tex]L[f(at)] = \int_0^\infty f(at)e^{-st}dt[/tex]
I was given a hint to let u = at --> dt = du/a so we have
[tex]L[f(at)] = \frac{1}{a}\int_0^\infty f(u)e^{-\frac{s}{a}u}\,du[/tex]
Now it looks like I am about done, but I am not sure how to proceed? I believe I now need to show that the if by definition
[tex]L[f(t)] = F(s) = \int_0^\infty f(t)e^{-st}dt[/tex]
then the integral
[tex]\int_0^\infty f(t)e^{-\frac{s}{a}t}\,dt = F(\frac{s}{a})[/tex]
Seems simple enough, but I am not sure how to show it.