MATH 20D Lecture 26

Theorem.

For .
Proof:

We make the subs. so when and .
Thus,

Example. Find the Laplace transform of the function $f(t)=\{\begin{array}{l}-2,t<1\\ 1,1\le t<3\\ -5,3\le t\end{array}$

We know that so

Let

Let , another type unit step function is given by
$$u_c(t)=u(t-c)=\begin{cases}0,t<c\1,t\geq c\end{cases}c\geq0$

Unit step functions can be used as an “on-off switch” to control the behavior of a given function over certain time intervals. In general, if is a piecewise function given by

Then can be expressed in terms of unit step functions as

Theorem

For , the Laplace transform of the unit step function is given by

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Example. Find the Laplace transform for the following functions

$u(t-2)e^{7t}$

for

$u(t-\pi /2)$ and $u(t-\pi /2)\sin (t-\pi /2)$

for

for

$u(t-1)(t^2+3t+2)$

for

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Example. Find the inverse Laplace transform for $Y(s)=e^{-3s}\frac{s+1}{(s-3)(s-5)}$.

Suggest the final answer is where
We find using partial fraction decomposition
Here,
So

This is not . This is actually
We do one final shift to find

So