# The laplace transform for ramp function with a time delay can be expressed as follows

The laplace transform for ramp function with a time delay can be expressed as follows
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Step 1
Let the time delay is T,
So the ramp function will be
$f\left(t\right)=\left(t-T\right)u\left(t-T\right)$
Where
$u\left(t-T\right)=\left\{\begin{array}{ll}1& t>T\\ 0& t\end{array}$
Step 2
Let the Laplace transform of f(t) is F(s)
So the Laplace transform is
$F\left(s\right)={\int }_{0}^{\mathrm{\infty }}f\left(t\right){e}^{-st}dt$
$⇒F\left(s\right)={\int }_{T}^{\mathrm{\infty }}T\left(t-T\right){e}^{-st}dt$
let
x=t-T
$⇒dx=dt$
So we have
$F\left(s\right)={\int }_{0}^{\mathrm{\infty }}x{e}^{-s\left(x+T\right)}dx$
$⇒F\left(s\right)={e}^{-sT}{\int }_{0}^{\mathrm{\infty }}x{e}^{-sx}dx$
$⇒F\left(s\right)=\frac{{e}^{-sT}}{{s}^{2}}$