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

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

2020-10-27
Step 1
Let the time delay is T,
So the ramp function will be
$$f(t)=(t-T)u(t-T)$$
Where
$$u(t-T)=\begin{cases}1 & t>T\\0 & t Step 2 Let the Laplace transform of f(t) is F(s) So the Laplace transform is \(F(s)=\int_0^\infty f(t)e^{-st}dt$$
$$\Rightarrow F(s)=\int_T^\infty T(t-T)e^{-st}dt$$
let
x=t-T
$$\Rightarrow dx=dt$$
So we have
$$F(s)=\int_0^\infty xe^{-s(x+T)}dx$$
$$\Rightarrow F(s)=e^{-sT}\int_0^\infty xe^{-sx}dx$$
$$\Rightarrow F(s)=\frac{e^{-sT}}{s^2}$$

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