What is Laplace: Definition and 1000 Discussions

Pierre-Simon, marquis de Laplace (; French: [pjɛʁ simɔ̃ laplas]; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.
Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries.
He was Napoleon's examiner when Napoleon attended the École Militaire in Paris in 1784.
Laplace became a count of the Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.

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  1. C

    Use the Shifting Theorem to find the Laplace transform

    For (b), I'm confused on the highlighted step. Does someone please explain to me how they got from the left to the right? Thanks!
  2. C

    Laplace transform of cosine squared function

    For part (b), I have tried finding the Laplace transform of via the convolution property of Laplace transform. My working is, ##L[\cos^2 (2t)] = L[\cos 2t] * L[\cos 2t]## ##L[\cos^2 (2t)] = \frac{s}{s^2 + 4} * \frac{s}{s^2 + 4}## ##\int_0^t \frac{s^2}{(s^2 + 4)^2} dt = \frac{ts^2}{(s^2 +...
  3. C

    Laplace transform of proper rational function

    For this problem (b), The solution is, However, I don't understand how they got their partial fractions here (Going from step 1 to 2). My attempt to convert into partial fractions is: ##\frac{2s + 1}{(s - 1)(s - 1)} = \frac{A(s - 1) + B(s - 1)}{(s - 1)(s - 1)}## Thus, ##2s + 1 = A(s - 1) +...
  4. P

    How to Simplify the Laplace Equation in Spherical Coordinates?

    I know what the Laplace operator is and I also looked up how f(r,θ,φ)=Rl(r)Ylm(θ,φ) is defined but I still could not solve the problem.
  5. 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...
  6. 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} \\...
  7. 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?
  8. 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}...
  9. P

    I Laplace equation with irregular boundaries

    Is there a way to solve Laplace’s Equation on irregular domains if the domain’s shape is given by a function for example a 2D parabolic plate. I keep seeing numerical methods but I want to know is there an ANALYTICAL method to solve it on an irregular domain. If there isn't are there approximate...
  10. M

    MacKay Textbook Example: Laplace Approximation

    Hi, I was attempting example 27.1 question from the book: 'Information Theory, Inference, and Learning Algorithms'. It is about the Laplace approximation. I was confused about part (b) of the question and wanted to check my method if possible. [EDIT]: The link to the book website (official) is...
  11. A

    Partial fraction decomposition with Laplace transformation in ODE

    Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
  12. matqkks

    What is the best way to introduce Laplace transforms for Engineers?

    Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function. How can Laplace transforms be introduced so that students are...
  13. matqkks

    MHB What is the best way to introduce Laplace transforms in an Engineering Mathematics course?

    Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function. How can Laplace transforms be introduced so that students are...
  14. 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##?
  15. A

    Engineering How would I solve this using Laplace transformation?

    Hello! Consider this transferfunction H(s); $$ H(s) =\frac{s-1}{1-2(s^2-s)-As-\frac{A}{2}} $$ Now I need to determine A (note that A is coming from R) so that the impulse response h(t) (so in time domain) so that it contains components with $$te^{at} \sigma(t) $$. Now I honestly really have...
  16. yucheng

    Confused about the nature of Laplace vs Poisson equation in BVP

    Hi! The problem clearly states that there is a surface charge density, which somehow gives rise to a potential. The author has solved the Laplace equation in cylindrical coordinates and applied the equation to the problem. So ##\nabla^2 V(r,\phi) = 0##, and ##V(a,\phi) = V_a(\phi)## (where...
  17. 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)})...
  18. 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^-##?
  19. L

    I Convergence of this Laplace transformation

    I have a f(t) that is, e^(-t) *sin(t), now I calculate the Laplace transformation, that is: X(s) = 1 / ( 1 + ( 1 + s)^2 ) (excuse me but Latex seems not run ). Now I imagine the plane with Re(s), Im(s) and the magnitude of X(s). If i take Re(s) = -1 and Im(s) = 0, I believe I have X(s) = 1 ( s...
  20. M

    MHB Model the situation with a Laplace room

    Hey! 😊 An ice cream parlour offers 12 different types of ice cream, including vanilla ice cream. There are 8 people passing by, each of whom chosses a ball of ice cream. Of course, the ice cream parlour has taken good precautions, so that there is enough ice cream from each variety. Model the...
  21. Safinaz

    B Does the Laplace operator equal the Del operator squared?

    Hello , The Laplace operator equals ## \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} ## so does it equal as well nable or Del operator squared ## \bigtriangledown^2## ? where ## \bigtriangledown =\frac{\partial}{\partial...
  22. L

    Is the Fourier Transform Correctly Applied in Solving This Laplace Equation?

    I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result: $$ \tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky} $$ However, I'm having some problems with the inverse transform: $$ \frac{1}{2\pi}\int_{-\infty}^\infty...
  23. J

    3D Laplace solution in Cylindrical Coordinates For a Hollow Cylindrical Tube

    Here is the initial problem and my attempt at getting Laplace solution. I get lost near the end and after some research, ended up with the Bessel equation and function. I don't completely understand what this is or even if this i the direction I go in. This is a supplemental thing that I want to...
  24. 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...
  25. H

    Laplace transforms for which value of s?

    I was wondering how you work out what values of s a Laplace transform exists? And what it actually means? The example given in class is an easy one and asks to calculate the Laplace transform of 3, = 3 * Laplace transform of 1 = 3 * 1/s. Showing this via the definition, where does the range of s...
  26. 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...
  27. 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'...
  28. 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?
  29. B

    Python Laplace approximation in Bayesian inference

    Hello everybody, I am working on a Python project in which I have to make Bayesian inference to estimate 4 or more parameters using MCMC. I also need to evaluate the evidence and I thought to do so through the Laplace approximation in n-dimensions: $$ E = P(x_0)2\pi^{n/2}|C|^{1/2} $$ Where C...
  30. 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...
  31. E

    B What is the Significance of the Laplace Operator in Vector Calculus?

    ##\frac {\partial \vec F} {\partial x} ## + ##\frac{\partial \vec F} {\partial y} ## = vector which gives me a direction of the greatest increase of the greatest increase of the function, where ##\vec F ## = gradient of the function. If I multiple the first by ##\hat i## and the second by ##\hat...
  32. jawad hussain

    Laplace Equation Numerical Solution

    I wonder how to incorporate point charge?
  33. docnet

    Radial solutions to the Laplace equation

    Part 1 $$\Delta u(x)=\Delta v(|x|)$$ Substitute $$|x|=r=\sqrt{\sum_{i=1}^n{x^2_i}}$$ $$u'(x)= v'(r)\frac{\sum_{i=1}^nx_i}{\sqrt{\sum_{i=1}^n{x^2_i}}}$$ $$u''(x)=v''(r)\frac{\sum_{i=1}^nx_i}{\sqrt{\sum_{i=1}^n{x^2_i}}}+v'(r)f(x)=v''(r)+v'(r)f(x)$$...
  34. docnet

    Prove the rotational invariance of the Laplace operator

    Hello, please lend me your wisdom. ##\Delta u=\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u## ##Rx=\left<r_{11}x_1+...r_{1n}x_n+...+r_{n1}x_1+...+r_{nn}x_n\right>## ##(\Delta u)(Rx)=(\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u)\left<r_{11}x_1+...r_{1n}x_n...
  35. 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 $$...
  36. rannasquaer

    MHB How to Solve Laplace Transforms with a Fractional Term?

    How to solve the transforms below \[ \mathscr{L}^{-1} \frac{a(s+2 \lambda)+b}{(s+ \lambda)^2- \omega^2} \]
  37. 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)).
  38. 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...
  39. 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?
  40. 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 =...
  41. 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...
  42. 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...
  43. 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...
  44. 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...
  45. 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?
  46. L

    Engineering Help with Homework: Solving a Math Formula

    i want to ask about my homework im not understand what to do with this formula :