What is Laplace transform: Definition and 776 Discussions

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable


{\displaystyle t}
(often time) to a function of a complex variable


{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral








{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}

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  1. Z

    Laplace transform of ##f(t)=(u(t)-u(t-2\pi))\sin{t}##

    I tried to solve this as follows $$f(t)=(u(t)-u(t-2\pi))\sin{t}$$ $$=u(t)\sin{t}-u(t-2\pi)\sin{t}$$ $$\mathcal{L}(f(t))=e^{0\cdot s}\mathcal{sin{t}}-e^{-2pi s}\mathcal{L}(\sin{(t+2\pi)})$$ $$=\frac{1-e^{-2\pi s}}{s^2+1}$$ where I used the fact that ##\sin{(t+2\pi)}=\sin{t}##. Then I looked...
  2. Z

    Partial fractions with complex linear terms

    I am interested specifically in solving this problem by factoring the quadratic term into complex linear factors. $$s^2+4=0$$ $$\implies s=\pm 2i$$ $$\frac{5s+6}{(s-2i)(s+2i)(s-2)}=\frac{A}{s-2i}+\frac{B}{s+2i}+\frac{C}{s-2}$$ We can solve for ##C## using the cover-up method with ##s=2## to...
  3. bnich

    I Getting zeros and poles for Laplace transform

    I'm following the intuition behind getting the zero's and poles of a damped cosine function with this video At around 11:50, he shows some graphics pertaining to multiplying the probing function with the impulse response, but the graphics don't seem correct. For example, in the B+B' graphic...
  4. P

    I On convolution theorem of Laplace transform: Schiff

    Here follows the theorem and proof: Questions: 1. I do not understand the following part "...and hence, in view of the preceding calculation, ##\int_0^\infty \int_0^\infty |e^{-st}f(\tau)g(t-\tau)|dtd\tau## converges". We know that ##\mathcal{L}\big(f(t)\big)## and...
  5. P

    I On Laplace transform of derivative

    The following three results are used in the proof of the theorem I have a question about. Now follows the theorem and its proof I have a question about. I do not understand why ##e^{-st}f(t)\rvert_0^\infty=-f(0)##. By Lemma 2, this is only possible if ##f## is of exponential type of order...
  6. Wrichik Basu

    Laplace transform vs phasor analysis in circuit analysis

    I recently acquainted myself with Laplace transform, and it appears that it has some relations with phasor analysis. This observation stems from the fact that while in Laplace transform, we have ##s = \sigma + j \omega## as the variable, in phasor analysis, we just use ##j\omega,## apparently...
  7. M

    I Laplace Transform of Sign() or sgn() functions

    Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity...
  8. M

    Engineering How to implement a transfer function in Simulink with variable coefficients?

    The implementations for the two filters in simulink are as follow: For the first filter: For the second one: The obtained results have values of 10^-12, while the expected results should be between 10^-3 - 10. Since it's the first time when I try t implement a tf with variable coefficients I...
  9. H

    Solving ##y'' - 5 y' - 6y = e^{3x}## using Laplace Transform

    We have to solve $$ \begin{align*} y'' - 5y' - 6y = e^{3x} \\ y(0) = 2,~~ y'(0) = 1 \\ \end{align*} $$ Applying Laplace Transform the equation $$ \begin{align*} L [ y''] - 5 L[y'] - 6 L[y] = L [ e^{3x} ] \\ s^2 Y(s) - \left( s y(0) + y'(0) \right) - 5s Y(s) + y(0) - 6 Y(s) = \frac{1}{s-3} \\...
  10. L

    A Applying the Laplace transform to solve Differential equations

    Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?
  11. greg_rack

    Check on proof for property of the Laplace transform

    Could someone check whether my proof for this simple theorem is correct? I get to the result, but with the feeling of having done something very wrong :) $$\mathcal{L} \{f(ct)\}=\int_{0}^{\infty}e^{-st}f(ct)dt \ \rightarrow ct=u, \ dt=\frac{1}{c}du, \ \mathcal{L}...
  12. H

    Mellin transform of Dirac delta function ##\delta(t-a)##

    Hi, I found Laplace transform of this Dirac delta function which is ##F(s) = e^{-st}## since ##\int_{\infty}^{-\infty} f(t) \delta (t-a) dt = f(a)## and that ##\delta(x) = 0## if ##x \neq 0## Then the Mellin transform ##f(t) = \frac{1}{2 \pi i} \int_{\gamma - i \omega}^{\gamma +i \omega}...
  13. L

    I Laplace transform of a simple equation (Simple question)

    Lets consider very simple equation ##x''(t)=0## for ##x(0)=0##, ##x'(0)=0##. By employing Laplace transform I will get s^2X(s)=0 where ##X(s)## is Laplace transform of ##x(t)##. Why then this is equivalent to X(s)=0 why we do not consider ##s=0##?
  14. 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)})...
  15. L

    A Laplace transform of derivatives

    I have a question regarding Laplace transforms of derivatives \mathcal{L}[f'(t)]=p\mathcal{L}[f(t)]−f(0^−) Can anyone explain me why ##0^-##?
  16. Joan Fernandez

    A Why is the MGF the Laplace transform?

    The Laplace transform gives information about the exponential components in a function, as well as oscillatory components. To do so there is a need for the complex plane (complex exponentials). I get why the MGF of a distribution is very useful (moment extraction and classification of the...
  17. A

    I Understanding the Laplace Transform of cos(t)/t

    So, I know the direct definition of the Laplace Transform: $$ \mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$ So when I plug in: $$\frac{\cos(t)}{t}$$ I get a divergent integral. however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29 is supposed to be the...
  18. H

    What is the Inverse Laplace Transform of e^(-sx^2/2)?

    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'...
  19. 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?
  20. willDavidson

    Laplace Transform Finding Open-Circuit Voltage

    I am interested in modeling a battery charging/discharging. I am starting off with a simple model using a voltage source in series with a parallel RC branch which is in series with a resistor. I will be measuring the open circuit voltage between the last series resistor and the bottom of the...
  21. patric44

    Does the Laplace Transform of e^(at)/t Exist?

    hi guys i am facing a little problem calculating this Laplace transform ## \mathscr{L}(\frac{e^{\alpha t}}{t})## , when calculate it using the method of the inverse Laplace transform its equal to $$ ln{\frac{1}{s-\alpha}}$$ but then when i try to use the theorem $$...
  22. rannasquaer

    MHB How to Solve Laplace Transforms with a Fractional Term?

    How to solve the transforms below \[ \mathscr{L}^{-1} \frac{a(s+2 \lambda)+b}{(s+ \lambda)^2- \omega^2} \]
  23. Frankenstein19

    I Laplace transform linearity problem

    I've included the problem statement and a bit about the function but my main issue is with the equation after "then" and the one with the red asterisk. I don't understand why the Laplace transform for a u(t)*e^(-t/4) isn't (1/s)*(1/(s+1/4)). The book I am reading says it's(1/(s+1/4)).
  24. 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...
  25. S

    Laplace transform of an ODE with a non-smooth forcing function

    Suppose I'm solving $$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?
  26. jisbon

    Engineering Nodal Analysis of this Circuit using the Laplace Transform

    Was just practicing some problems on the Fundamentals of Electric Circuits, and came across this question. I understand I will have to transform to the s domain circuit, which looks something like this: Then doing nodal analysis, I will get the following for the first segement (10/s-V1)/1 =...
  27. 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...
  28. PainterGuy

    Solving an ODE with the Laplace transform

    Hi again, The previous problem was done using y′′(t)+2y′(t)+10y(t)=10 with with intial condition y(0⁻)=0. In the following case, I'm using an initial condition and setting the right hand side equal to zero. Find y(t) for the following differential equation with intial condition y(0⁻)=4...
  29. PainterGuy

    MATLAB Finding an inverse Fourier transform using the Laplace transform

    Hi, This thread is an extension of this discussion where @DrClaude helped me. I thought that it'd be better to separate this question. I couldn't find any other way to post my work other than as images so if any of the embedded images are not clear, just click on them. It'd make them clearer...
  30. PainterGuy

    I Laplace transform of an expression using transform tables

    Hi, I 'm trying to find the Laplace transform of the following expression. I used the following conversion formulas. I think "1" is equivalent to unit step function who Laplace transform is 1/s. I ended up with the following final Laplace transform. Is my final result correct? Thank you...
  31. L

    Engineering Laplace transform of the given circuit

    Hello i have an assignment. From given circuit i need to find s domain and inverse them back to t domain. can you help me by explain this circuit?
  32. engnrshyckh

    B What Is the Correct Inverse Laplace Transform of 1/s(s²+w²)?

    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.
  33. P

    MHB Solving Integral Equation w/ Laplace Transform - Abdullah

    We would need to recognise that the integral in the equation is a convolution integral, which has Laplace Transform: $\displaystyle \mathcal{L}\,\left\{ \int_0^t{ f\left( u \right) \,g\left( t - u \right) \,\mathrm{d}u } \right\} = F\left( s \right) \,G\left( s \right) $. In this case...
  34. P

    MHB Seth's question via email about a Laplace Transform

    Since this is of the form $\displaystyle \frac{f\left( t \right)}{t} $ we should use $\displaystyle \mathcal{L}\,\left\{ \frac{f\left( t \right) }{t} \right\} = \int_s^{\infty}{F\left( u \right) \,\mathrm{d}u } $. Here $\displaystyle f\left( t \right) = \cosh{\left( 4\,t \right) } - 1 $ and so...
  35. P

    MHB Jun's question via email about Laplace Transform

    Upon taking the Laplace Transform of the equation we have $\displaystyle \begin{align*} s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 4\,Y\left( s \right) &= -\frac{8\,\mathrm{e}^{-6\,s}}{s} \\ s^2 \,Y\left( s \right) - 2\,s - 0 + 4\,Y\left( s \right) &=...
  36. P

    MHB Dharshan's question via email about a Laplace Transform

    $\displaystyle \begin{align*} \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \sinh{ \left( 11\,t \right) } \right\} &= \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \cdot \frac{1}{2} \left( \mathrm{e}^{11\,t} - \mathrm{e}^{-11\,t} \right) \right\} \\ &= \frac{5}{2} \,\mathcal{L} \left\{...
  37. Jason-Li

    Comp Sci Laplace Transform of the input portion of this circuit

    So I have completed (a) as this (original on the left): I have then went onto (b) and I have equated T(s)=Z(s) as follows: and due to hence Does this look correct to you smarter people? Thanks in advance! All replies are welcome :)
  38. 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 ?
  39. G

    MHB Solve integral with laplace transform

    So the task is to solve the following integral with laplace transform. Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)). What I am unsure about is what happens with the integral part and how do we inpret the resulting expression? What will it result...
  40. L

    A Laplace transform in spherical coordinates

    Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...
  41. PainterGuy

    I Solving a differential equation using Laplace transform

    Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...
  42. E

    Help Proving a Complex Laplace Transform

    So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd...
  43. SamRoss

    I Why is the Laplace transform unchanged when t is replaced with -t?

    In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as... $${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$ The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...
  44. cnh1995

    Physical Significance of the Laplace Transform

    I have used Laplace transform during my EE studies to solve differential equations and in control system analysis, but we were taught that as a tool kit to make the math easier. The physical meaning was never explained. I know basic time and frequency domain concepts (thanks to Fourier series)...
  45. C

    Engineering Advanced Circuits, Laplace Transform, Find Initial Conditions

    Vo(S) = [ N(s)Vi(s) + (- s2 + s - 2) ] / s3 + s2 + 1 ; can ignore (-s^2 + s - 2). From relevant equations: Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d3Vo(t)/dt3) + (d2Vo(t)/dt2) + Vo(t) = N(t)(dvi)/dt L[vi(t)] = t to s domain: [s3Vo(s) - s2Vo(0-) - SV'o(0-) - Vo''(0-)]Vo(s) + s2 - SVo -...
  46. L

    Laplace transform of sin(ωt)/[1+cos^2(ωt)]

    Homework Statement L{sin(ωt)/[1+cos^2(ωt)]} = Homework Equations d {arctan[cos(ωt)]} /dt = - ω•sin(ωt)/[1+cos^2(ωt)] The Attempt at a Solution ∫e^(-st)•[sin(ωt)/(1+cos²(ωt)] dt = -(1/ω)•∫e^(-st)•{arctan[cos(ωt)]}' dt = = (integrating by parts and taking Re(s) > 0) = = π/(4ω) -(s/ω)•∫...
  47. dRic2

    Simplifying Laplace Transform of Cosine with Angular Frequency and Phase Shift

    Homework Statement I have to find the L-transform of ##f(x) = cos(\omega t + \phi)## Homework Equations . The Attempt at a Solution The straightforward approach is to write ##cos(\omega t + \phi)## as ##cos(\omega t)cos(\phi) - sin(\omega t)sin(\phi)## and it becomes: $$Lf(s) = \frac {s...
  48. thoraxepi

    I Transfer Function relating momentum and force

    Hey all, I hope this is the correct forum section to post this in. I heard about this problem from a youtube video but I've not been able to simulate it because the video was meant only for an introduction into PID control. Here's the problem: A remote control helicopter is hovering just...
  49. C

    Solving a 2D PDE using the Fourier Transform

    Homework Statement Solve the following partial differential equation , using Fourier Transform: Given the following: And a initial condition: Homework EquationsThe Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...