The problem arose when I was trying to compute a Laplace transform of a functional equation.
I have a known nonlinear function, $f(x)$, where $x=g(t)$ is an unknown function of time. I wish to compute $L(f(g(t))$, at least to the extent where I can obtain useful information about the equation in question. My strategy was to expand $f(x)$ as a Taylor series about $x=g(t)=0$, and attempt a Laplace transform term-by-term. The specific point doesn't actually matter, but I wanted to try the most simple case first. The result is a polynomial equation of $g(t)$. However,
$$ L\left(\sum_{i=0}^\infty A_i \right) = \sum_{i=0}^\infty L(A_i) $$
Only works if the Laplace transformed series is uniformly convergent. For example, if you were to have an explicit function, such as $e^{-t^2}$, you could compute the Laplace transform of the Taylor series term-by-term, and show the Laplace transformed series is divergent, hence the above approach will fail. However, the 2-sided Laplace transform for this is well known and is computed in any college level probability textbook.
Now, I do know if $f(g(t))$ is bounded (there might be one or two other conditions I am forgetting off the top of my head), then the Laplace transform does exist even if it cannot be computed analytically.
But do any of these rules apply for functional equations? Better yet, I cant find any good references on this topic, and if someone could direct me to resources I would really appreciate that. Here is another example:
$$ TSE(cos(x))|(x=0) = \sum_{i=0}^\infty -1^i\frac{x^{2i}}{(2i)!} $$
However, the next example looks like garbage to me, though I cant think of why it is necessarily wrong (aside from the fact that $sin(t)$ has zeros for $t = n \pi$):
$$ TSE(cos(sin(t)))|(sin(t)=0) = \sum_{i=0}^\infty -1^i\frac{sin(t)^{2i}}{(2i)!} $$
My math background beyond college level engineering is cobbled together from self study for my research in grad school, so I know a bit about Lie algebra, differential geometry, symplectic geometry, stochastic processes, etc, but I have no formal education in real analysis or functional analysis. I studied adaptive and geometric control theory, specifically, which is already too mathematical for most engineers but probably not mathematical enough for real math majors. Hence, there are probably some fundamental gaps in my knowledge that need to be closed and I appreciate any insight/references.
edit: made TSE about $x$, $sin(t)$ look neater.