Let $a,b\in \mathbb{N}$, $a>b$. Does the Laplace transform of the following exist? $$f(t)=(1-H_{a} (t)) \exp{[(-1)^b (t-a)^2+(t-a)^{2021}]}$$
Should Ι distinguish the cases for $b$?
- $b$ is odd
- $b$ is even
2026-04-24 02:26:50.1776997610
Does the Laplace transform of $f(t)=(1-H_{a} (t)) \exp{[(-1)^b (t-a)^2+(t-a)^{2021}]}$ exist?
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$$\mathcal{L} (f(t)) = \int_0^a e^{(-1)^b (t - a)^2 + (t - a)^{2021}} \cdot e^{-st} \: dt$$ This is an integral of a continuous function over the closed interval $[0, a]$. So it exists. The fun part is evaluating it in closed form.