# 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:

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{\displaystyle {\mathcal {L}}\{f\}(s)={\mathcal {L}}\{f(t)\}(s)=F(s),}
where

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{\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.

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1. ### Inverse Laplace transform

\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)})...
2. ### Inverse Laplace transform

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'...
3. ### MHB Inverse laplace transform pf infinite product

I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?
4. ### Why is the heaviside function in the inverse Laplace transform of 1?

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...
5. ### Ladder-like bandpass filter theoretical analysis problem

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...
6. ### I Inverse Laplace transform of a rational function

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...
7. ### B Inverse Laplace transform

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.
8. ### MHB Laplace / inverse laplace transform

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 ?

30. ### Inverse Laplace Transform with e^{a s}

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...
31. ### MHB Inverse Laplace Transform 2

And also: y+2y=2(1-e^-2t) Y(0)=0 y¨-2y+y = t+e^t y(0)=1 and y(0)=0 Please help me out here folks ;)
32. ### MHB Inverse Laplace Transform 1

Hi. Im 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:)
33. ### Inverse laplace transform without partial fractions

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?
34. ### Inverse Laplace Transform

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.
35. ### Inverse Laplace transform. Bromwitch integral

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...
36. ### Inverse Laplace Transform

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...
37. ### Tricky inverse Laplace transform

<< 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)) *...
38. ### Inverse Laplace transform with p^-1 and exponential

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...
39. ### Inverse laplace transform (polynomial division? Complex roots?)

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...
40. ### Inverse Laplace transform where e^(st)F(s) is entire

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...
41. ### Inverse Laplace transform

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)} =...
42. ### Help please -- inverse Laplace transform of 1/(x^2+1)^2

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 :)
43. ### Inverse Laplace transform

Homework Statement Ds + E / (s^2 +1)^2 Homework Equations The Attempt at a Solution Ds / (s^2 +1) + E / (s^2 +1) D[s/(s^2 + 1)^2] + E [1 / (s^2 + 1)^2]
44. ### MHB Inverse laplace transform II

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!

48. ### MHB Inverse Laplace transform question

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 \[...
49. ### Inverse Laplace transform

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?
50. ### Inverse Laplace Transform Help

Homework Statement Is there a way to evaluate L^{-1}(\frac{F(s)}{s + a})? I'm sure if it can be evaluate. Homework Equations The Attempt at a Solution