Question

If f(t) satisfies those conditions f(t) is even function which has the property of f(t) = f(-t) Laplace transform of f(t) is F(s) then Laplace transform of f(at) is F(s/lal)/lal for any non-zero constant a Prove this - How about odd function f(t) which has the property of f(-t) -f(t) ?

How to prove this? Laplace transform

Answer 1

a)

e st f(at) dt L(f(at))

Consider 2 cases:

We substitute and our equation becomes

du at u dt =

L(f(at)) ef(u) du a

But as
est f(t) dt F(s)
we have
e st/a f(t) dt F(s/a)

So that

F L(f(at))

Now the case in which case we substitute
b a
so that

est fbt) dt L(f(at)) est f(at) dt

Using even property, we get $L(f(at))=\int_{0}^{\infty}e^{-st}f(bt)\,dt=\frac{1}{b}F\left ( \frac{s}{b} \right )$

That is, for we have $L(f(at))=\frac{1}{(-a)}F\left ( \frac{s}{(-a)} \right )$

Combining these we get $L(f(at))=\frac{1}{|a|}F\left ( \frac{s}{|a|} \right )$

b) If we have an odd function, we get:

then
F L(f(at))
as the equation is unaffected

If then we get

est fbt) dt L(f(at)) est f(at) dt

Which is $L(f(at))=-\int_{0}^{\infty}e^{-st}f(-bt)\,dt$ and so

L(f(at)) -F -a) (-a)

So that $L(f(at))=\text{sgn}(a)\frac{1}{|a|}F\left(\frac{s}{|a|} \right )$