I've recently been introduced to Laplace transforms, and my understanding so far is that it's a continuous analogue to a Summation of a power series, that maps injectively a function $f(t)$ to another function of a new variable $s$, $F(s)$.
My question though is that if we take $$\int_0 ^\infty a(t)x ^t dt$$And then make the substitution $-s = ln(t)$ and that the coeffecients $a(t)$ are generated by the function $f(t)$ we get: $$\int_0 ^\infty f(t)e^{-st}dt.$$
My confusion comes from looking up and finding that $s$ is apparently of the form $\sigma + i\omega$, but given that $-s$ is $\ln(x)$ and the limits are $0$ to $\infty$, that surely means that $-s \in R$? Or have I made a mistake in my reasoning somewhere.
The second part of my question is with regards to $s$ again, in that from what I've seen of Fourier transforms [hasn't come in syllabus yet] it just seems like a Laplace transform with purely imaginary $s$. What is / was the mathematical motivation for this?