I posted this query within the math stackexchange boards, and different extra standard however much less skilled math associated communities, however didn’t get and suggestions. I hope by posting right here I can get a solution to my query. Under is a precise copy of my unique submit:

The issue arose once I was attempting to compute a Laplace rework of a purposeful equation.

I’ve a identified nonlinear operate, $f(x)$, the place $x=g(t)$ is an unknown operate of time. I want to compute $L(f(g(t))$, at the least to the extent the place I can acquire helpful details about the equation in query. My technique was to broaden $f(x)$ as a Taylor sequence about $x=g(t)=0$, and try a Laplace rework term-by-term. The particular level would not really matter, however I needed to attempt the most straightforward case first. The result’s a polynomial equation of $g(t)$. Nevertheless,

$$

Lleft(sum_{i=0}^infty A_i proper) = sum_{i=0}^infty L(A_i)

$$

Solely works if the Laplace remodeled sequence is uniformly convergent. For instance, when you have been to have an specific operate, akin to $e^{-t^2}$, you possibly can compute the Laplace rework of the Taylor sequence term-by-term, and present the Laplace remodeled sequence is divergent, therefore the above strategy will fail. Nevertheless, the 2-sided Laplace rework for that is well-known and is computed in any school stage chance textbook.

Now, I do know if $f(g(t))$ is bounded (there may be one or two different circumstances I’m forgetting off the highest of my head), then the Laplace rework does exist even when it can’t be computed analytically.

However do any of those guidelines apply for purposeful equations? Higher but, I cant discover any good references on this subject, and if somebody may direct me to assets I might actually recognize that. Right here is one other instance:

$$

TSE(cos(x))|(x=0) = sum_{i=0}^infty -1^ifrac{x^{2i}}{(2i)!}

$$

Nevertheless, the subsequent instance seems to be like rubbish to me, although I cant consider why it’s essentially incorrect (except for the truth that $sin(t)$ has zeros for $t = n pi$):

$$

TSE(cos(sin(t)))|(sin(t)=0) = sum_{i=0}^infty -1^ifrac{sin(t)^{2i}}{(2i)!}

$$

My math background past school stage engineering is cobbled collectively from self examine for my analysis in grad faculty, so I do know a bit about Lie algebra, differential geometry, symplectic geometry, stochastic processes, and so forth, however I’ve no formal schooling in actual evaluation or purposeful evaluation. I studied adaptive and geometric management principle, particularly, which is already too mathematical for many engineers however most likely not mathematical sufficient for actual math majors. Therefore, there are most likely some elementary gaps in my data that must be closed and I recognize any perception/references.

edit: made TSE about $x$, $sin(t)$ look neater.