Is the Laplace transform the continuous version of the infinite power series?
$$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$
I learned this by watching this video lecture: https://www.youtube.com/watch?v=sZ2qulI6GEk.
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Requires more information to answer by a larger number of people. For example, $a : X \to Y$ is a map, what are $X$ and $Y$ in the above? What variable are you summing over? Write $t=0$ below the sum symbol. Define $a(t)$ as well. – Daniel Donnelly Jul 15 '14 at 22:58
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1I guess the question I have for you is, what would you want to be true for the answer to be "Yes?" They seem alike in some ways, and very different in others. Why wouldn't it be more akin to $\sum_{k=0}^\infty a_ne^{-ns}$? – Thomas Andrews Jul 15 '14 at 23:11
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In the lecture $s = -\ln(x)$. Can I use this always when dealint with Laplace transforms? That's my question. – mrk Jul 15 '14 at 23:15
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https://math.stackexchange.com/a/3430967/525644 This answer is from a different vantage point – Aravindh Vasu Nov 27 '19 at 14:50