structural:summarization, linguistic:paraphrasing
Unilateral Laplace transform

The Laplace transform of the Heaviside step function is a meromorphic function. Using the unilateral Laplace transform we have: H ^ (s) = lim N → ∞ ∫ 0 N e − s x H (x) d x = lim N → ∞ ∫ 0 N e − s x d x = 1 s {\displaystyle {\begin{aligned}{\hat {H}}(s)&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}H(x)\,dx\\&=\lim _{N\to \infty }\int _{0}^{N}e^{-sx}\,dx\\&={\frac {1}{s}}\end{aligned}}}

When the bilateral transform is used, the integral can be split in two parts and the result will be the same.