Laplace transform: The meaning of s and its link to Fourier Transforms.

manpreet Best Answer 2 years ago

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)f(t) to another function of a new variable ss, F(s)F(s).

My question though is that if we take

And then make the substitution −s=ln(t)−s=ln(t) and that the coeffecients a(t)a(t) are generated by the function f(t)f(t) we get:

My confusion comes from looking up and finding that ss is apparently of the form σ+iωσ+iω, but given that −s−s is ln(x)ln⁡(x) and the limits are