Step Functions

Used in Differential Equations that have discontinuities on the right hand side

Heaviside Step Function

u (t) = {\begin{cases} 0, t < 0 \\ 1, t \geq 0 \end{cases}

For a constant $a$ :

u (t - a) = {\begin{cases} 0, t - a < 0 \\ 1, t - a \geq 0 \end{cases} = {\begin{cases} 0, t < a \\ 1, t \geq a \end{cases}

Laplace Transforms of the Heaviside Function

L {u (t - a)} = \int_{0}^{\infty} e^{- s t} u (t - a) d t = \frac{e^{- s a}}{s}