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
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(
s
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=


L


{
f
(
t
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}
(
s
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=
F
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s
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,


{\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.

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  1. L

    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. H

    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. M

    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. B

    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. R

    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. R

    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. engnrshyckh

    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. G

    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 ?
  9. evinda

    MHB Inverse Laplace transform

    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}...
  10. J

    Inverse Laplace Transform

    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...
  11. Cocoleia

    Laplace & Inverse Laplace transforms

    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...
  12. Vitani11

    Inverse Laplace transform for an irreducible quadratic?

    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...
  13. D

    Inverting Shifted Laplace function

    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...
  14. cg78ithaca

    A Inverse Laplace transform of a piecewise defined function

    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...
  15. cg78ithaca

    A Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)

    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...
  16. K

    How to find the Inverse Laplace Transform of this function?

    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?
  17. C

    Inverse Laplace transform

    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...
  18. C

    Inverse Laplace transform

    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...
  19. H

    A tricky inverse Laplace transform

    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...
  20. I

    Inverse Laplace transform of 1/s

    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)...
  21. D

    Inverse Laplace transform for 1/(350+s) * X(s)

    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...
  22. P

    MHB Henry's question via email about an Inverse Laplace Transform

    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...
  23. R

    Inverse Laplace Transform

    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...
  24. P

    MHB Emad's question via email about Inverse Laplace Transform

    The closest Inverse Laplace Transform from my table is $\displaystyle \begin{align*} \mathcal{L}^{-1}\,\left\{ \frac{2\,a\,s\,\omega}{\left( s^2 + \omega ^2 - a^2 \right) ^2 + 4\,a^2\,\omega ^2 } \right\} = \sin{ \left( \omega \, t \right) } \sinh{ \left( a \, t \right) } \end{align*}$ so we...
  25. Mark Brewer

    Inverse Laplace Transform

    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)...
  26. jdawg

    Inverse Laplace Transform

    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...
  27. Italo Campoli

    Inverse Laplace Transform of a fractional F(s)

    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...
  28. P

    MHB Johnsy's question over Facebook about an Inverse Laplace Transform

    To start with, apply partial fractions... $\displaystyle \begin{align*} \frac{A\,s + B}{ s^2 + 9} + \frac{C\,s + D}{ \left( s^2 + 9 \right) ^2 } &\equiv \frac{ 4s^3 + 5s^2 + 57s + 45}{ \left( s^2 + 9 \right) ^2 } \\ \left( A\,s + B \right) \left( s^2 + 9 \right) + C\,s + D &\equiv 4s^3 + 5s^2 +...
  29. R

    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...
  30. K

    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 ;)
  31. K

    MHB Inverse Laplace Transform 1

    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:)
  32. S

    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?
  33. R

    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.
  34. L

    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...
  35. _N3WTON_

    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...
  36. M

    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)) *...
  37. kewei chen

    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...
  38. M

    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...
  39. Q

    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...
  40. M

    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)} =...
  41. N

    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 :)
  42. K

    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]
  43. D

    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!
  44. D

    MHB Inverse laplace transform of a function

    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}\} =...
  45. M

    Inverse Laplace Transform and Court

    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...
  46. P

    MHB Douglas' question via email about Inverse Laplace Transform

    To start with, let's work on the Partial Fraction decomposition. $\displaystyle \begin{align*} \frac{A\,s + B}{s^2 - 16s +128} + \frac{C\,s + D}{s^2 + 16s + 128} &\equiv \frac{3s}{s^4 + 16\,384} \\ \frac{ \left( A\,s + B \right) \left( s^2 + 16s + 128 \right) + \left( C\,s + D \right) \left(...
  47. D

    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 \[...
  48. B

    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?
  49. G

    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
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