📂Odinary Differential Equations

Laplace Transform of Constant Functions

Formulas¹

$$\mathcal{L} \left\{ 1 \right\} = \dfrac{1}{s},\quad s>0$$

Derivation

$$\begin{align*} \mathcal{L}\left\{ 1 \right\} &= \int _{0}^\infty e^{-st} \cdot 1 dt \\ &= \lim \limits_{A \to \infty} \left[ -\dfrac{e^{-st}}{s} \right]_{0}^A \\ &= \lim \limits_{A \to \infty} \left[ -\dfrac{e^{-sA}}{s} +\dfrac{e^{-0t}}{s} \right] \\ &= \dfrac{1}{s} \end{align*}$$

Since it must follow $\lim \limits_{A \to \infty}\dfrac{e^{-sA}}{s}=0$,² the condition that $s>0$ is added.

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See Also

• Table of Laplace Transforms