I don't think there is a Laplace transformation for either the
function 1/x or ln x and I think I have a proof of this. Does
this check out?
Let 1/x = 1/(1 - y) These functions are in principle the same,
i.e. f(x) = f(1 - y)
Also, 1/(1 - y) = 1 + y + y2 + y3 +
y4 .....
(for |y| < 1)
Hence L(1/(1 - y) ) = L(1 + y + y2 .......)
Letting L(y) = F(s) we get
L(1 + y + y2 +.....) = 0!/s + 1!/s2 +
2!/s3 + 3!/s4 .............
This is the sum from 1 to infinity of (n-1)!/sn
However, if we apply the ratio test, we obtain the result that
this series has a radius of convergence of infinity, and so it
cannot converge, and so therefore there cannot be a Laplace
transformation for 1/x.
The argument for ln x is similar; we let x = 1-y and take the
power series for ln(1-y)
= 1 + y + y2 /2 + y3 /3 + y4 /4
+......
We then carry out the Laplace transform on that series again, and
find that it also has a radius of convergence of infinity, and so
cannot have a Laplace transform. I am worried about a couple of
things about this proof though, for instance it only applies when
mod y< 1; could there be some range of values for which there
is a Laplace transform for 1/x or ln x?