The First Shifting Theorem is a fundamental property of Laplace transforms that states if you shift a function in the time domain by a constant, its Laplace transform is multiplied by an exponential factor in the s-domain. Specifically, if you have a function $$f(t)$$ and you shift it to $$f(t - a)$$ for $$t \geq a$$, then the Laplace transform of this shifted function relates to the original by the equation $$\mathcal{L}\{f(t - a)u(t - a)\} = e^{-as}F(s)$$, where $$F(s)$$ is the Laplace transform of $$f(t)$$. This theorem is important for analyzing systems in engineering and physics where delays or shifts are present.
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