Question

Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = [infinity] e−stf(t) dt 0 is said to be the Laplace transform of f, provided that the integral converges. Find ℒ{f(t)}. (Write your answer as a function of s.) f(t) = −1, 0 ≤ t < 1 1, t ≥ 1 ℒ{f(t)} = (s > 0)

Answer 1

$f(t)=\begin{cases}-1&\text{for }0\le t<1\\1&\text{for }t\ge1\end{cases}$

Write $f(t)$ in terms of the unit step function $u(t)$:

$f(t)=-1(u(t)-u(t-1))+1u(t-1)=2u(t-1)-u(t)$

where

$u(t)=\begin{cases}1&\text{for }t\ge0\\0&\text{for }t<0\end{cases}$

Then the Laplace transform of $f(t)$, denoted $F(s)$, is

$F(s)=\displaystyle\int_0^\infty f(t)e^{-st}\,\mathrm dt$

$F(s)=\displaystyle2\int_1^\infty e^{-st}\,\mathrm dt-\int_0^\infty e^{-st}\,\mathrm dt$

$\implies F(s)=\dfrac{2e^{-s}}s-\dfrac1s=\dfrac{2e^{-s}-1}s$