Unit Step Function

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Subsection 6.5.1 Unit Step Function

With the help of unit step functions, we can find the inverse transform of such functions which can not be determined with the previously explained methods. The unit step functions $u(t-a)$ is defined as

\begin{equation*} u(t-a) = \begin{cases} 0, & \text{where} \quad t \lt a\\ 1, & \text{where} \quad t\geq a \end{cases} \end{equation*}

and $a\geq 0\text{.}$ The unit step function can be visulized as figure Figure 6.5.1.

Subsubsection 6.5.1.1 Laplace Transform of Unit Step Function

\begin{equation*} L[u(t-a)] =\frac{e^{-as}}{s} \end{equation*}

Proof.

\begin{equation*} \mathscr{L}[u(t-a)] =\int\limits_{0}^{\infty} e^{-st}u(t-a)\,dt \end{equation*}

\begin{equation*} =\int\limits_{0}^{a} e^{-st}.0\,dt + \int\limits_{a}^{\infty} e^{-st}.1\,dt = 0+\left[\frac{e^{-st}}{-s}\right]_{a}^{\infty}= \frac{e^{-as}}{s}. \end{equation*}

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