~~not trying to reinvent the Laplace transform, but just an exploration into these particular series and integrals~~
Current answers don't fully address the 5 questions, so any new ideas or suggestions would be much appreciated. Thanks for the help!
The Taylor Series for $e^x$ is $$\sum \frac{x^n}{n!}$$ Now isn’t this just a discrete sum of functions? What if I use integrals to make a "continuous" version of the Taylor series? Following that motivation, I came up with $$E(x)=\int_{-\infty}^\infty \frac{x^n}{G(n)}dn$$ where $G(n)=\Gamma(n+1)=n!$. Since $\frac{x^n}{G(n)}$ goes to zero as $n\to -1$, the integral just becomes $$E(x)=\int_{-1}^\infty \frac{x^n}{G(n)}dn$$ I graphed it on Desmos and it looked like this, with the green dotted line being $E(x)$:
After that, I was like "wow, nice! I wonder if other functions work too”. Naturally, I moved on to $\sin(x)$, which has power series $$\sum \frac{(-1)^nx^{2n+1}}{(2n+1)!}$$ Unfortunately, this power series is more complicated because of the $(-1)^n$ term. The first thought I came up with is that $\cos(\pi n)$ could be the continuous version of that term. So, one such “continuized” version of the Taylor series for $\sin x$ would be $$S(x)=\int_{-1}^\infty \frac{\cos(\pi n) x^{2n+1}}{G(2n+1)}dn$$ Of course, that's kind of arbitrary, so I did use two other functions: $$c_1(x)=\cos^6\left(\frac{\pi x}{2}\right)-\sin^6\left(\frac{\pi x}{2}\right)$$ which is more "triangular", and $$c_2(x)=2\left(1-\sin^6\left(\frac{\pi x}{2}\right)\right)^6-1$$ which is more "square". You can see all three of these functions in orange. I made three integrals with each of the three functions, with the dotted green line with the one with $\cos(\pi x)$. I had to multiply it by a factor of $2$ to get it right, interestingly enough. In both integrals, I made the lower bound a bit higher to avoid crashing my computer and the higher bound low enough to make little to no difference.
(accessible here: https://www.desmos.com/calculator/eesis3ykai, though it may take a while to load)
The first one has already been addressed here: The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$, so my only contribution is the nice graph. However, the $\sin x$ and $\cos x$ ones I found far more fascinating. Hence, I have a few questions:
(1) does $S(x)$ actually converge to $\sin(x)$?
(2) can we expect this "integral-Taylor series" to work on a lot of other functions? Is there some general result?
(3) why does $\cos(\pi x)$ work the best compared to the triangle and square waves? Why did the rectangular wave fail so badly?
(4) why does the integral have to be stretched by a factor of two, when the $e^x$ integral didn’t have to be?
(5) is this approximation for these functions useful? Can this method be applied elsewhere? Is there any use to this outside just "ooh look at this neat graph"?
More cool stuff: I did the same thing with $\cos x$, and I got similar results (the bounds get shifted a bit though):
$$C(x)=\int_{-0.5}^\infty \frac{\cos(\pi n) x^{2n}}{G(2n)}dn$$
(which is accessible here: https://www.desmos.com/calculator/ctjqdxuw0h)
which also has the strange multiplicative factor of $2$, and the peculiar favorability to $(-1)^n \approx \cos(\pi n)$. So my above 5 questions still stand.