What is Inverse laplace transform: Definition and 165 Discussions
In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property:
L
{
f
}
(
s
)
=
L
{
f
(
t
)
}
(
s
)
=
F
(
s
)
,
{\displaystyle {\mathcal {L}}\{f\}(s)={\mathcal {L}}\{f(t)\}(s)=F(s),}
where
L
{\displaystyle {\mathcal {L}}}
denotes the Laplace transform.
It can be proven that, if a function F(s) has the inverse Laplace transform f(t), then f(t) is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem.The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems.
\mathcal{L}^{-1}[\frac{e^{-5s}}{s^2-4}]=Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=2]+Res[e^{-5s}\frac{1}{s^2-4}e^{st},s=-2]
From that I am getting
f(t)=\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)}. And this is not correct. Result should be
f(t)=\theta(t-5)(\frac{1}{4}e^{2(t-5)}-\frac{1}{4}e^{-2(t-5)})...
My attempt at finding this was via convolution theorem, where we take F(s) = 1/s^2 and G(s) = e^(-sx^2/2). Then to use convolution we need to find the inverses of those transforms. From a table of Laplace transforms we know that f(t) = t. But I am sort of struggling with e^(-sx^2/2). My 'guess'...
Homework Statement:: Why is the heaviside function in the inverse laplace transform of 1?
Relevant Equations:: N/A
This is a small segment of a larger problem I've been working on, and in my book it gives the transform of 1 as 1/s and vice versa. But as I've looked online for help in figuring...
I consider the band-pass filter of the following configuration (the ##u_m## is a voltage controlled voltage source):
The transfer function is
$$K_1(p)=\hat{U}_o(p) = \frac{p}{RC(p+1/RC)^2} = \frac{\omega_c p}{(p+\omega_c)^2}, \quad \omega_c=\frac{1}{RC}.\qquad (1)$$
Now I connect ##n## such...
I struggle to find an appropriate inverse Laplace transform of the following
$$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$
WolframAlpha gives as an answer
$$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$
which...
I used partial fraction method first as:
1/s(s^2+w^2)=A/s+Bs+C/(s^2+w^2)
I found A=1/w^2
B=-1
C=0
1/s(s^2+w^2)=1/sw^2- s/s^2 +w^2
Taking invers laplace i get
1/w2 - coswt
But the ans is not correct kindly help.
Problem: Find a (limited?) solution to the diff eq.
At the end of the solution, when you transform \frac{-1}{s+1} + \frac{2}{s-3}
why doesn't it become -e^{-t} + 2e^{3t} , t>0 ?
Hello! (Wave)
I want to find $f(t)$ if its Laplace transform is $F(s)=\frac{1}{s(s^2+1)}$.
We use the following formula, right?
$$f(t)=\frac{1}{2 \pi i} \lim_{T \to +\infty} \int_{a-iT}^{a+iT} e^{st} F(s) ds$$
But how can we calculate the integral $\int_{a-iT}^{a+iT} e^{st}...
Homework Statement
Y=(8s-4)/(s²-4)
Homework EquationsThe Attempt at a Solution
I rearranged the right side as:
8*(s/(s²-2²))-2*(2/(s²-2²))
Using the Laplace transform chart given in the class I was able to identify these as the transforms of hyperbolic sine and hyperbolic cosine making the...
Homework Statement
I am given this equation:
and asked to solve using Laplace transforms
The Attempt at a Solution
This is what I did:
This seemed logical to me, I used partial fractions and it stayed pretty simple.
This is what the solutions my prof posted do:
Is my answer equivalent...
Homework Statement
I have to take the inverse Laplace of this function (xoms+bxo)/(ms2+bs+k) this can not be broken into partial fractions because it just gives me the same thing I started with. How is this done? This is coming from the laplace of the position function for a harmonic oscillator...
Homework Statement
A beam is supported at one end, as shown in the diagram (PROBLEM 11 page 281 of Lea, 159 of the course pack). A block of mass M and length l is placed on the beam, as shown. Write down the known conditions at x = 0. Use the Laplace transform to solve for the beam...
I understand the conditions for the existence of the inverse Laplace transforms are
$$\lim_{s\to\infty}F(s) = 0$$
and
$$
\lim_{s\to\infty}(sF(s))<\infty.
$$
I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as
$$F(s) =\begin{cases} 1-s...
This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple.
I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...
Homework Statement
ƒ(s) = 1/((1-exp(-s))*(1+s))
Homework EquationsThe Attempt at a Solution
I know the solution is periodic but how to obtain the t-domain function?
Homework Statement
Determine the inverse Laplace transform
Homework Equations
3s+9.
(s+3)^2+7
The Attempt at a Solution
[/B]
Hi iam new to the forum and still unsure how to make the equations the correct format so hope you can understand what I have typed.
I have Tried to Convert...
Homework Statement
Determine the inverse Laplace transform
Homework Equations
3s+9/(s+3)^2+7
The Attempt at a Solution
Converted to 3s+9/s^2+6s+16 to try and use the partial fractions method but getting nowhere.
I'm Not sure if Iam making the question more difficult, can't seem to put the...
Homework Statement
I want to invert a function from Laplace transform space to normal space.
Homework Equations
In Laplace transform space, the function takes the form $$ \bar f (s) = \frac{\exp\left[ x (-a +\sqrt{a^2+ b +c s} )\right]}{-a +\sqrt{a^2+ b +c s}}.
$$
Here, ##s## is the Laplace...
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
The left hand side (red box) is the data sheet provided to us in the exam. The right hand side (blue box) is Wolfram Alpha. The data sheet says that the inverse Laplace transform of 1/s is equal to u(t) (i.e. the unit step)...
Hi, everyone, the question is as below:
Find the inverse Laplace transform to 1/(350+s) * X(s). 's' is the Laplace variable and 'X(s)' is also a variable.
I inverted 1/(350+s) and X(s) separately and multiplied them together directly. But this seems not giving me the correct answer. Could...
It's not entirely obvious what to do with this question, as the denominator does not easily factorise. However, if we realize that $\displaystyle \begin{align*} s^4 + 40\,000 = \left( s^2 \right) ^2 + 200^2 \end{align*}$ it's possible to do a sneaky completion of the square...
$\displaystyle...
Homework Statement
Given the Laplace transform
$$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$
by using the complex inversion formula compute the inverse Laplace transform, ##f(t),## for the following regions of convergence:
(i) ##Re(s)<-2;##
(ii) ##-2<Re(s)<0;##
(iii) ##Re(s)>0.##
Homework Equations...
Homework Statement
L-1{(2s2+3)/(s2+3s-4)2}
The Attempt at a Solution
I factored the denominator
f(t)=(2s2+3)/((s-1)(s+4))2
now I've tried partial fractions to get
(2s2+3)/((s-1)(s+4))2 = A/(s-1)2 + B(s+4)2
(2s2+3)=A(s+4)2 + B(s-1)2
by substitution, s=1 and s=-4
5=A(25)
A=1/5
35=B(25)...
Homework Statement
L-1{3/s1/2}
Homework EquationsThe Attempt at a Solution
3L-1{1/s1/2}
3L-1{(1/sqrt(π))(sqrt(π)/(sqrt(s))}
3/(sqrt(π))L-1{(sqrt(π))/(sqrt(s))}
3/(sqrt(π))(1/(sqrt(t))
This is what I got from the solution for this problem. What tipped them off to multiply by sqrt(π)? And...
Homework Statement
[/B]
Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor
2. The attempt at a solution
tryed at first with partial fractions but that didnt got me anywhere, i know i could use tables at the 2nd fraction i got as...
Homework Statement
How can I take the Inverse Laplace Transform of $F(s) = \frac{d}{ds}\left(\frac{1-e^{5s}}{s}\right)$?
I have tried going with inverse of the derivative and convolution (even tried evaluating the derivative and go from there) but although I can get to some results none of them...
Hi.
I`m new here and I need some help with Inverse Laplace Transform: f(t)=5+3t+e^3t g(t)=(t+1)u(t-2) g(t)=(t^2-9t+20)u(t-5) and Laplace Transform: F(s)=1/(s+2)^5 F(s)= 2s^2+10/s(s^2+2s+10) G(S)=2s/s^2+4e^-sso if anywone can please help me:)
Homework Statement
take inverse laplace of:
6/[s^4(s-2)^2]
Homework Equations
6/[s^4(s-2)^2]
The Attempt at a Solution
I used partial fractions. I was wondering if It could be manipulated to where I could use the laplace table?
Homework Statement
Find H(s) = \frac{Y(s)}{X(s)}
\frac {d^2y(t)}{dt^2} + a\frac {dy(t)}{dt} = x(t) + by(t)
Homework EquationsThe Attempt at a Solution
[s^2 + as - b] Y(s) = X(s)
H(s) = \frac{1}{s^2+as-b}
I assume the inverse is a sign or a cosine but unsure which one.
Inverse Laplace transform
\mathcal{L}^{-1}[F(p)]=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}F(s)e^{st}dp=f(t)
Question if we integrate along a straight line in complex plane where axis are Re(p), Im(p), why we integrate from c-i \ínfty to c+\infty? So my question is, because Im(p) are also...
Homework Statement
Division by s Equals integration by t:
For this problem use the following property (see relevant equations) to find the inverse transform of the given function: F(s) = \frac{1}{s(s-1)}
Homework Equations
L^{-1}(\frac{F(s)}{s}) = \int_{0}^{t} f(\tau)\,d \tau
The Attempt...
<< Moderator Note -- thread moved to the Homework Help forums >>
I'm stuck on a problem, and I'm in serious need of help.
I) Problem:
Find the solution to f (t) = 2 \int^t_0 f'(u) sin 3 (t-u) \ du + 2 cos (3t) .
Also find f (0) .II) Solution, so far:
F(s) = 2 (s F(s) - f(0)) *...
Hello everyone, I have spend whole day but still not figure out an inverse Laplace transform. Hope someone can help me. The question is in the attachment. I'm trying to extract u^2/4D^2 out the bracket to match the standard inverse table, but it seems difficult to deal with the square root...
Homework Statement
Decide the inverse laplace transform of the problem below:
F(s)= \frac{4s-5}{s^2-4s+8}
You're allowed to use s shifting.
Homework Equations
The Attempt at a Solution
By looking at the denominator, I see that it might be factorized easily, so I try that...
Heya folks,
I'm currently pondering how to decide whether a function has an inverse Laplace transform or not. In particular I am considering the function e^(-is), which I am pretty sure does not have an inverse Laplace transform. My reasoning is that when calculating the inverse by the Bromwich...
Homework Statement
Find the inverse Laplace transform of the expression:
F(S) = \frac{3s+5}{s^2 +9}
Homework Equations
The Attempt at a Solution
From general Laplace transforms, I see a pattern with laplace transforming sin(t) and cos(t) because:
L{sin(t)+cos(t)} =...
Homework Statement
Hi.
I need help to resolve the inverse laplace transform of {1/((x^2)+1)^2}2. The attempt at a solution
I have tried to do:
{(1/((x^2)+1) * (1/((x^2)+1)}
then, convolution, sen x
But, isn't working
Thanks for your help :)
before I go to bed(it's 11:30pm in my place), here is the last problem that I need help with
find the inverse Laplace Transform
$\frac{4s-2}{s^2-6s+18}$
the denominator is a non-factorable quadratic. I don't know what to do.
thanks!
find the inverse Laplace of the ff:
1. $\frac{n\pi L}{L^2s^2+n^2 \pi^{2}}$
2. $\frac{18s-12}{9s^2-1}$
for the 2nd prob
I did partial fractions
$\frac{18s-12}{9s^2-1}=\frac{9}{3s+1}-\frac{3}{3s-1}$
$\mathscr{L}^{-1}\{\frac{18s-12}{9s^2-1}\} =...
Homework Statement
I had a question in my midterm, it was to find inverse laplace tansform of:
(4s+5) / (s^2 + 5s + 18.5)
Where ^ denotes power.
Homework Equations
The Attempt at a Solution
My answer was to find the complex roots of equation (s^2 + 5s + 18.5) , by them...
With a Laplace transform, we can remember common set ups; for example,
\[
\mathcal{L}\{e^{-at}\} = \frac{1}{s + a}.
\]
When it comes to the inverse Laplace transform, I can only find the tables to remember in a book. However, if we go back to the Laplace transform, we can always do
\[...
Hi,
I would like to find the inverse Laplace transform for
11/(s^2+16)^2
I have tried to expand it using the following partial fraction decomp to find the constants and take the inverse Laplace but this did not work
C1(s)+ C2/(s^2+16) + C3(s)+C4/(s^2+16)^2
Does anyone have any suggestions?