Heaviside Function

u (t - τ) = {\begin{cases} 0 & if & t < τ \\ 1 & if & t \geq τ \end{cases}

Screenshot 2023-04-26 at 7.15.05 PM.png

Theorem (Translation in $t$ )

Let $F (s) = L {f (t)}$ and assume $τ > 0$ .

Then

L {u (t - τ) f (t - τ)} = e^{- τ s} F (s)

and so $u (t - τ) f (t - τ) = L^{- 1} {e^{- τ s} F (s)}$

Special Case:

f (t) = 1 ⟹ L {u (t - τ)} = \frac{e^{- τ s}}{s}