Laplace transform limits?

matematikuvol

How we get relation
[tex]\lim_{t\to 0}f(t)=\lim_{p\to \infty}pF(p)[/tex]?

Where ##\mathcal{L}\{f\}=F##.

mathman

pF(p) = p∫e^-ptf(t)dt. Integrate by parts with du = pe^-ptdt and v = f(t). Then (assuming f(t) reasonable) let p -> ∞ and you get the desired result.

matematikuvol

I saw also assymptotics relation
##\lim_{t \to \infty}f(t)=\lim_{p\to 0}pF(p)##
when that relation is valid?

mathman

I am not familiar with this. However for most cases, both sides = 0.

matematikuvol

For ##1## both sides are equal ##1##. ##lim_{t\to \infty}1=1=lim_{p\to 0}p\frac{1}{p}=1##. I think that is correct only if both limits converge.