Does the Laplace Transform of e^(at)/t Exist?

[patric44]

Homework Statement

find Laplace transform of e^(at)/t

Relevant Equations

L(e^at/t)

hi guys
i am facing a little problem calculating this Laplace transform $\mathscr{L}(\frac{e^{\alpha t}}{t})$ , when calculate it using the method of the inverse Laplace transform its equal to
$$ ln{\frac{1}{s-\alpha}}$$
but then when i try to use the theorem
$$ \mathscr{L}(\frac{f(t)}{t})=\int_{s}^{\infty}F(s)ds=\int_{s}^{\infty}\mathscr{L}(f(t))ds = \int_{s}^{\infty}\frac{1}{s-\alpha}ds$$
$$=lim_{s→∞}(s-\alpha)-ln(|s-\alpha|)$$
it seems that there is a term that will blow up to infinity!
what i am missing here?!

 

[Office_Shredder]

I think you should start by writing down the integral that is the definition of the laplace transform, and ask yourself if it exists.

 

[patric44]

I think you should start by writing down the integral that is the definition of the laplace transform, and ask yourself if it exists.

i know that it will lead me to the exponential integral function Ei(x) , but its easy to show that the $L^{-1}(ln\frac{1}{s-\alpha}) = e^{\alpha t}/t$
my question is about this particular theorem why it doesn't hold here ?

 

[Mark44]

i know that it will lead me to the exponential integral function Ei(x) , but its easy to show that the $L^{-1}(ln\frac{1}{s-\alpha}) = e^{\alpha t}/t$
my question is about this particular theorem why it doesn't hold here ?

The assumption in the Laplace transform is that $f$ in the integral $\int_0^{\infty} f(t)e^{-st}dt$ must be one that is well behaved, in the sense that it is at least conditionally convergent at infinity.

 

[Delta2]

I think the laplace transform of this function simply does not exist because the integral $$\int_0^{+\infty}\frac{e^{-(s-a)t}}{t}dt$$ does not exist, because for example the integral $$\int_0^1\frac{e^{-(s-a)t}}{t}dt$$ does not exist and the "rest" integral $$\int_1^{+\infty}\frac{e^{-(s-a)t}}{t}dt$$ is simply positive.

That theorem doesn't hold because the theorem has some requirements (at least according to my book which is written in Greek by Greek authors) and one of the requirements is the limit $$\lim_{t\to 0}\frac{f(t)}{t}$$ to be a real number, which in this case doesn't hold.