As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've copied below.

Consider the Laplace transform $\mathfrak{L}\{f\}:=\int_0^\infty e^{-st}f(t)~dt$ on a smooth function $f$ such that there exists some $s_0\in\mathbb{C}$ such that $e^{-s_0t}f(t)\to0$ as $t\to\infty$ (I believe it is sufficient to require $f$ is of exponential type). By integration by parts, it follows $$\mathfrak{L}\{f\}=s^{-1}f(0)+s^{-2}f'(0)+s^{-3}f''(0)+\dots=s^{-1}\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}$$

In other words, the Laplace transform behaves like a map from $f$ to the generating function of $(f^{(n)}(0))_{n\in\mathbb{N}}$ at $t=0$.

For example, consider a straightforward linear constant-coefficient ODE in $f$, say $f''-2f'+f=0$. 'Applying' the Laplace transform, we get: $$s^{-1}\sum_{n=0}^\infty f^{(n+2)}(0)~s^{-n}-2s^{-1}\sum_{n=1}^\infty f^{(n+1)}(0)~s^{-n}+s^{-1}\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}=0\\s^{-1}\left(s^2\sum_{n=2}^\infty f^{(n)}(0)~s^{-n}\right)-2s^{-1}\left(s\sum_{n=1}^\infty f^{(n)}(0)~s^{-n}\right)+s^{-1}\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}=0\\s\left(\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}-f(0)-f'(0)s^{-1}\right)-2\left(\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}-f(0)\right)+s^{-1}\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}=0\\(s^2-2s+1)s^{-1}\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}=sf(0)+f'(0)+2f(0)\\s^{-1}\sum_{n=0}^\infty f^{(n)}(0)~s^{-n}=\frac{sf(0)+f'(0)+2f(0)}{s^2-2s+1}$$

This so far agrees with the elementary result found using the basic properties of the Laplace transform and, in fact, with the standard method of solving recurrence relations using generating functions.

Indeed, it seems as though the Laplace transform is simply the method of generating functions applied to the sequence $(f^{(n)}(0))_{n\in\mathbb{N}}$ (over which the ODE reduces to a recurrence relation). However it seems almost necessary that for this interpretation to work, the function's behavior over the domain of interest must be fully described by its derivatives at $t=0$, i.e. $f$ must be analytic.

What can be concluded from the above? Is this a valuable approach to looking at the Laplace transform?