TITLE: Laplace transform of $1/t$
QUESTION [13 upvotes]: Does the laplace transform of $1/t$ exist? If yes, how do we calculate it? Putting it in $$\int_0^\infty (e^{-st}/t) dt$$ won't solve.
Is there any other way? If not, why?
Thanks!

REPLY [4 votes]: You can actually simplify it further by substituting $st = x$, so you'll get
$$L \left(\frac 1 t \right) = \int \limits _0 ^\infty \frac {\Bbb e ^{-x}} x \ \Bbb d x$$
which is a divergent integral. In other words, the transform doesn't converge for any value of $s$.