# Use the definition of Laplace Transforms to find Lleft{f(t)right} f(t)=begin{cases}-1 & 0leq t <11 & tgeq 1end{cases} Then rewrite f(t) as a sum of step functions, u_c(t), and show that by taking Laplace transforms, this yields the same answer as your direct computation.

Question
Laplace transform
Use the definition of Laplace Transforms to find $$L\left\{f(t)\right\}$$
$$f(t)=\begin{cases}-1 & 0\leq t <1\\1 & t\geq 1\end{cases}$$</span>
Then rewrite f(t) as a sum of step functions, $$u_c(t)$$, and show that by taking Laplace transforms, this yields the same answer as your direct computation.

2021-02-13
Step 1
Consider the given function:
$$f(t)=\begin{cases}-1 & 0\leq t <1\\1 & t\geq 1\end{cases}$$</span>
Step 2
Now, by definition of Laplace Transform:
$$L\left\{f(t)\right\}=\int_0^\infty e^{-st}f(t)dt$$
$$=\int_0^1e^{-st}f(t)dt+\int_1^\infty e^{-st}f(t)dt$$
$$=\int_0^1e^{-st}(-1)dt+\int_1^\infty e^{-st}(1)dt$$
$$=\left[\frac{e^{-st}}{s}\right]_0^1+\left[\frac{e^{-st}}{-s}\right]_1^\infty$$
$$=\left(\frac{e^{-s}}{s}-\frac{1}{s}\right)+\left(0+\frac{e^{-s}}{s}\right)$$
$$L\left\{f(t)\right\}=\frac{2e^{-s}}{s}-\frac{1}{s}$$
Step 3
Now, Rewrite f(t) as the sum of step functions, $$u_c(t)$$:
$$f(t)=f(t)=\begin{cases}-1 & 0\leq t <1\\1 & t\geq 1\end{cases}$$</span>
$$=(-1)(u_0,1(t))+(1)(u_1(t))$$
$$=(-1)(u_0(t)-u_1(t))+(1)(u_1(t))$$
$$=-u_0(t)+2u_1(t)$$
Step 4 Now, use the following formula for Laplace transform of step function:
$$L\left\{u_c(t)\right\}=\frac{e^{-cs}}{s}$$
Step 5
Thus, now take Laplace transform:
$$f(t)=-u_0(t)+2u_1(t)$$
$$L\left\{f(t)\right\}=L\left\{-u_0(t)\right\}+L\left\{2u_1(t)\right\}$$
$$=-\frac{e^{-(0)s}}{s}+2\frac{e^{-(1)s}}{s}$$
$$L\left\{f(t)\right\}=-\frac{1}{s}+2\frac{e^{-s}}{s}$$
Step 6
Thus, from equation (1) and (2), it can be seen that the Laplace transform of f(t) that is obtained from both the methods is same that is :
$$L\left\{f(t)\right\}=2\frac{e^{-s}}{s}-\frac{1}{s}$$

### Relevant Questions

Given the function $$\begin{cases}e^{-t}& \text{if } 0\leq t<2\\ 0&\text{if } 2\leq t\end{cases}$$
Express f(t) in terms of the shifted unit step function u(t -a)
F(t) - ?
Now find the Laplace transform F(s) of f(t)
F(s) - ?
Let f(t) be a function on $$\displaystyle{\left[{0},\infty\right)}$$. The Laplace transform of fis the function F defined by the integral $$\displaystyle{F}{\left({s}\right)}={\int_{{0}}^{\infty}}{e}^{{-{s}{t}}} f{{\left({t}\right)}}{\left.{d}{t}\right.}$$ . Use this definition to determine the Laplace transform of the following function.
$$\displaystyle f{{\left({t}\right)}}={\left\lbrace\begin{matrix}{1}-{t}&{0}<{t}<{1}\\{0}&{1}<{t}\end{matrix}\right.}$$
Let x(t) be the solution of the initial-value problem
(a) Find the Laplace transform F(s) of the forcing f(t).
(b) Find the Laplace transform X(s) of the solution x(t).
$$x"+8x'+20x=f(t)$$
$$x(0)=-3$$
$$x'(0)=5$$
$$\text{where the forcing } f(t) \text{ is given by }$$
$$f(t) = \begin{cases} t^2 & \quad \text{for } 0\leq t<2 ,\\ 4e^{2-t} & \quad \text{for } 2\leq t < \infty . \end{cases}$$
Find the Laplace transform of the given function
$$\begin{cases}t & 0,4\leq t<\infty \\0 & 4\leq t<\infty \end{cases}$$
$$L\left\{f(t)\right\} - ?$$
Find the Laplace transforms of the following time functions.
Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables.
a)$$f(t)=1+2t$$ b)$$f(t) =\sin \omega t \text{Hint: Use Euler’s relationship, } \sin\omega t = \frac{e^(j\omega t)-e^(-j\omega t)}{2j}$$
c)$$f(t)=\sin(2t)+2\cos(2t)+e^{-t}\sin(2t)$$
Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: $$(1) {1,i,-i}. (2) {-1+4i,-1-4i}. (3) {-1}. (4)$$ The empty diagram.
(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by $$w(t) = u(t)e^{-\frac{t}{2}} \sin(\frac{3t}{2})$$
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?
Existence of Laplace Transform
Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not)
a) $$f(t)=t^2\sin(\omega t)$$
b) $$f(t)=e^{t^2}\sin(\omega t)$$
Find the inverse Laplace transform $$f{{\left({t}\right)}}={L}^{ -{{1}}}{\left\lbrace{F}{\left({s}\right)}\right\rbrace}$$ of each of the following functions.
$${\left({i}\right)}{F}{\left({s}\right)}=\frac{{{2}{s}+{1}}}{{{s}^{2}-{2}{s}+{1}}}$$
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
$${\left({i}{i}\right)}{F}{\left({s}\right)}=\frac{{{3}{s}+{2}}}{{{s}^{2}-{3}{s}+{2}}}$$
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
$${\left({i}{i}{i}\right)}{F}{\left({s}\right)}=\frac{{{3}{s}^{2}+{4}}}{{{\left({s}^{2}+{1}\right)}{\left({s}-{1}\right)}}}$$
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
$$\begin{cases}t & 0\leq t<1\\ e^t & t\geq1 \end{cases}$$
$$L(f(t))=\int_0^1te^{-st}dt+\int_1^\infty e^{-(s+1)t}dt$$