# 4.3 Finding the inverse laplace transform

 Page 1 / 3

## Using transform tables

The inverse Laplace transform, given by

$x\left(t\right)=\frac{1}{2\pi j}{\int }_{\sigma -j\infty }^{\sigma +j\infty }X\left(s\right){e}^{st}ds$

can be found by directly evaluating the above integral. However since this requires a background in the theory of complex variables, which is beyond the scope of this book, we will not be directly evaluating the inverse Laplace transform. Instead, we will utilize the Laplace transform pairs and properties . Consider the following examples:

Example 3.1 Find the inverse Laplace transform of

$X\left(s\right)=\frac{{e}^{-10s}}{s+5}$

By looking at the table of Laplace transform properties we find that multiplication by ${e}^{-10s}$ corresponds to a time delay of 10 sec. Then from the table of Laplace transform pairs , we see that

$\frac{1}{s+5}$

corresponds to the Laplace transform of the exponential signal ${e}^{-5t}u\left(t\right)$ . Therefore we must have

$x\left(t\right)={e}^{-5\left(t-10\right)}u\left(t-10\right)$

Example 3.2 Find the inverse Laplace transform of

$X\left(s\right)=\frac{1}{{\left(s+2\right)}^{2}}$

First we note that from the table of Laplace transform pairs , the Laplace transform of $tu\left(t\right)$ is

$\frac{1}{{s}^{2}}$

Then using the $s$ -shift property in the table of Laplace transform properties gives

$x\left(t\right)=t{e}^{-2t}u\left(t\right)$

Also, the same answer may be arrived at by combining the Laplace transform of ${e}^{-2t}u\left(t\right)$ with the multiplication by $t$ property.

## Partial fraction expansions

Partial fraction expansions are useful when we can express the Laplace transform in the form of a rational function ,

$\begin{array}{cc}\hfill X\left(s\right)& =\frac{{b}_{q}{s}^{q}+{b}_{q-1}{s}^{q-1}+\cdots +{b}_{1}s+{b}_{0}}{{a}_{p}{s}^{p}+{a}_{p-1}{s}^{p-1}+\cdots +{a}_{1}s+{a}_{0}}\hfill \\ & =\frac{B\left(s\right)}{A\left(s\right)}\hfill \end{array}$

A rational function is a ratio of two polynomials. The numerator polynomial $B\left(s\right)$ has order $q$ , i.e., the largest power of $s$ in this polynomial is $q$ , while the denominator polynomial has order $p$ . The partial fraction expansion also requires that the Laplace transform be a proper rational function, which means that $q . Since $B\left(s\right)$ and $A\left(s\right)$ can be factored, we can write

$X\left(s\right)=\frac{\left(s-{\beta }_{1}\right)\left(s-{\beta }_{2}\right)\cdots \left(s-{\beta }_{q}\right)}{\left(s-{\alpha }_{1}\right)\left(s-{\alpha }_{2}\right)\cdots \left(s-{\alpha }_{p}\right)}$

The ${\beta }_{i},i=1,2,...,q$ are the roots of $B\left(s\right)$ , and are called the zeros of $X\left(s\right)$ . The roots of $A\left(s\right)$ , are ${\alpha }_{i},i=1,...,p$ and are called the poles of $X\left(s\right)$ . If we evaluate $X\left(s\right)$ at one of the zeros we get $X\left({\beta }_{i}\right)=0,i=1,...,q$ . Similarly, evaluating $X\left(s\right)$ at a pole gives The actual sign would need to be evaluated at some value of $s$ that is sufficiently close to the pole. $X\left({\alpha }_{i}\right)=±\infty ,i=1,...,p$ . The partial fraction expansion of a Laplace transform will usually involve relatively simple terms whose inverse Laplace transforms can be easily determined from a table of Laplace transforms. We must consider several different cases which depend on whether the poles are distinct.

## Distinct Poles:

When all of the poles are distinct (i.e. ${\alpha }_{i}\ne {\alpha }_{j},i\ne j$ ) then we can use the following partial fraction expansion:

$X\left(s\right)=\frac{{A}_{1}}{s-{\alpha }_{1}}+\frac{{A}_{2}}{s-{\alpha }_{2}}+\cdots +\frac{{A}_{p}}{s-{\alpha }_{p}}$

The coefficients, ${A}_{i},i=1,...,p$ can then be found using the following formula

${A}_{i}={\left(X,\left(s\right),\left(s-{\alpha }_{i}\right)|}_{s={\alpha }_{i}},i=1,...,p$

Equation [link] is easily derived by clearing fractions in [link] . The inverse Fourier transform of $X\left(s\right)$ can then be easily found since each of the terms in the right-hand side of [link] is the Laplace transform of an exponential signal. This method is called the cover up method .

Example 3.3 Find the inverse Laplace transform of

$\begin{array}{cc}\hfill X\left(s\right)& =\frac{2s-10}{{s}^{2}+3s+2}\hfill \\ & =\frac{2s-10}{\left(s+1\right)\left(s+2\right)}\hfill \end{array}$

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!