Trying to find the Laplace transform of \(\displaystyle{\frac{{{\cos{{t}}}}}{{{t}}}}\) It

Mary Bates 2022-04-08 Answered
Trying to find the Laplace transform of
costt
It comes out as infinity, but that doesn't make any sense.
Does this mean that this function doesn't have a Laplace transform or is something wrong here?
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Answers (1)

Jax Burns
Answered 2022-04-09 Author has 13 answers
Well, we are trying to find (solving a more general problem):
Lt[cos(nt)t](s)=0cos(nt)texp(st)dx
Using the 'frequency-domain integration' property of the Laplace transform, we can write:
Lt[cos(nt)t](s)=sLt[cos(nt)](σ)dσ
Using the table of selected Laplace transforms, we find:
Lt[cos(nt)](σ)=σσ2+n2
So, we get:
Lt[cos(nt)t](s)=sσσ2+n2dσ
Let's substitute u=σ2+n2 , so we get:
Lt[cos(nt)t](s)=12limϵs2+n2ϵ2+n21udu=12limϵ[ln|u|]s2+n2ϵ2+n2=
12limϵ(ln|ϵ2+n2|ln|s2+n2|)=12limϵ|ϵ2+n2s2+n2|
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