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. S

    Inverse laplace transform

    Homework Statement Derive an equation to represent the current flowing in an RLC circuit where R=12 L=1.2 C=30microfrards with an impulse voltage input. Assume zero conditions when the switch is closed. Homework Equations The Attempt at a Solution I have used the voltage to equal...
  2. M

    Using Laplace transform to solve this (simple) PDE

    Homework Statement Solve \frac{\partial^2 v}{\partial x^2} = \frac{\partial^2 v}{\partial t^2}, x > 0, t > 0 subject to v(x,0) = 0 v_t(x,0) = 0 v(0,t) = f(t) and where v is bounded for all x > 0, by taking Laplace transforms with respect to t. Homework Equations The Attempt at...
  3. M

    Laplace equation in N dimensional space

    Is it possible to obtain a general expression for the solution of laplace equation that is valid in an euclidean space with an arbitrary dimension ?
  4. R

    What is the Correct Way to Take the Inverse Laplace Transform?

    According to the Wikipedia page, the inverse Laplace transform is f(x) = \frac{1}{2 \pi i} \lim_{y\rightarrow \infty} \int_{x_0-iy}^{x_0+iy} F(s')e^{s'x}ds' Something seems wrong though. If I were to take the Laplace transform this equation, I should get F(s) coming out of the right hand side...
  5. S

    What is the Inverse Laplace Transform of (1/s) * (√s-1)/(√s+1)?

    Homework Statement Find the Inverse Laplace Transform of \frac{1}{s}*\frac{\sqrt{s}-1}{\sqrt{s}+1} The Attempt at a Solution for this question i found the singularities to be at 0 and when s = 1. (as the sqrt of 1 is ± 1) there is also a branch point that runs from 0→-∞. so if you...
  6. S

    Inverse Laplace Transform and Branch Points

    Homework Statement Find the inverse Laplace Transform of \frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1} Homework Equations The complex inversion formula (well known) The Attempt at a Solution The first thing is finding singularities and branch points and so on. From the \frac{1}{s}...
  7. P

    Inverse Laplace Transform: Solve Homework Equation

    Homework Statement Find the inverse laplace transform of the equation listed: (\frac{1-e^{-sa}}{a})*(\frac{1}{s^2}-\frac{RC}{s}+\frac{sRC+R^2\frac{C}{L}-1}{s^2+s\frac{R}{L}+\frac{1}{LC}}) Homework Equations a, R, C, and L are constants. The Attempt at a Solution Completely...
  8. T

    Laplace transform of convolution with derivative in it

    Homework Statement Hi, I am wondering how to Laplace transform this expression f(t)=\int^{\tau}_{0} g(\tau)f'(t-\tau)d\tau or more precisely f(t)=\int^{\tau}_{0} sin(8\tau)f'(t-\tau)d\tau The f'(t-\tau) gets me confused. Homework Equations \int^{\tau}_{0}...
  9. A

    Laplace Transformation Question

    Homework Statement Use Laplace transformations to solve: Y''+tY'+2Y=0, Y(0)=0, Y'(0)=1 Homework Equations Y''==>s^{2}y-sY(0)-Y'(0)=s^{2}y-1 tY'==>-(y+sy') 2Y==>2y The Attempt at a Solution The question felt pretty straightforward till I hit a rock.s^{2}y-1+y-sy'=0...
  10. P

    Simplifying Inverse Laplace Function

    Homework Statement Any way to reduce this function for simplification? (Trying to find the reverse Laplace function) I found this function after doing a circuit analysis on a transmission line. \frac{s^2RC+sL}{s^2RC+sL+R} Homework Equations noneThe Attempt at a Solution I couldn't...
  11. M

    Engineering Solving a circuit using laplace transform

    For the circuit below, find the Va(s) What I have done so far: I attempted to solve using the node method when the switch is closed (t > 0) (va - v0)/R1 + C2dva/dt + va/R2 = 0 After a series of calculations I got Va(s) = (V0/C2R1)*(1/s+(1/C2R1) + (1/C2R2)) What I am unsure of is if my first...
  12. C

    Laplace equations w/ discont. forcing functions

    Homework Statement y'' + 3y' + 2y = u_2(t); y(0)=0, y'(0)=1Homework Equations L\{u_c(t)h(t-c)\}=e^{-cs}H(s)The Attempt at a Solution I've actually tried two laplace equation with discontinuous forcing function problems. I get answers somewhat close to the back of the book, but still off. Maybe...
  13. C

    Laplace Transform of a Product

    I am given a function f(t) with it's corresponding Laplace Transform in the Frequency Domain (F(s)). I'm having a hard time wrapping my head around the product of say, L{cos(t)*f(t)}. The * is multiplication and not convolution. Must I do the integration for the Laplace transform by hand...
  14. D

    Verify that the function U is a solution for Laplace Equation.

    Verify that the function U = (x^2 + y^2 + z^2)^(-1/2) is a solution of the three-dimensional Laplace equation Uxx + Uyy + Uzz = 0. First I solved for the partial derivative Uxx, Ux = 2x(-1/2)(x^2 + y^2 + z^2)^(-3/2) = -x(x^2 + y^2 + z^2)^(-3/2) Uxx = -(x^2 + y^2 + z^2)^(-3/2) +...
  15. M

    Find a solution with Laplace

    Homework Statement Determine the solution for y^{''}+81y=81U(t-\frac{\pi }{2}) when \left\{y(0)=12,y'(0)=18\right\}U(t) is the unit step function The Attempt at a Solution Laplacetransforming : s^{2}Y(s)-sy(0)-y'(0)+81Y(s)=\frac{81e^{\frac{-\pi }{2}}}{s} With given data the equation...
  16. reddvoid

    How Is the ROC Right of the Rightmost Pole When There's a Pole at -2?

    in this question x(t) is right sided signal so ROC should be to the right of right most pole but here they've given ROC is Re(s) >-3 but there is a pole @ -2 how is this ?
  17. Q

    How to Compute the Laplace Transform of f(t-b)

    Homework Statement is it possible to find the laplace transform of f(t-b) ? i don't know if its possible, i am just trying. The Attempt at a Solution so, where integral from 0 to infinity, \int f(t-b) e-stdt let t-b = z =\intf(z) e-s(z+b)dz =\intf(z) e-sze-sbdz =e-sb f(s)...
  18. J

    Using Taylor Polynomial for Laplace Transforms

    Ive attached the problem and my work in the pic. Questions: Am I even applying the taylor polynomial the correct way? (I never learned taylor series, but I was supposed to be taught in the pre-requisite class) Am I suppose to plug in c=4? I am not so sure about how the U4(t) works...
  19. C

    Proof needed of link between 2 equations using Laplace Transforms

    Homework Statement Hello all. First of all I should mention that this is not a homework problem. Rather, during workplace research I have come across the following two equations in a journal paper. Here, the authors state that one equation can be shown to be the Laplace Transform of the other...
  20. C

    Inverse laplace tables, where am I going wrong?

    Homework Statement Find the inverse Laplace transform of \frac{2s+1}{s^2-2s+2}Homework Equations The Attempt at a Solution \begin{align*} L^{-1}\{ \frac{2s+1}{s^2-2s+2} \} &= 2 L^{-1}\{ \frac{s}{(s-1)^2+1} \} + L^{-1}\{ \frac{1}{(s-1)^2+1} \} \\ &= 2 L^{-1}\{ \frac{s}{(s-1)^2+1} \} + e^t...
  21. S

    Inverse Laplace Transform Equation

    Good day to all, I encounter this expression in analyzing my equation after transform it using Laplace Transform, to get the answer I have to invert it back, I have no idea on how to find its inversion. [text]-\text{Cosh}\left[\sqrt{2 s+s^2} x_0\right]+\text{Cosh}\left[s h_0+\sqrt{2 s+s^2}...
  22. P

    Find the inverse Laplace transform

    Find the inverse Laplace transform $F(s)=\frac{4}{s^4+4}$ I tried factoring out the solution, but run into the problem with the imaginary numbers and am still stuck with the s^2+2j, which I have to factor out once more, and that's where the problem gets even messier. What do I do?
  23. S

    Inital value problem, Laplace transform

    Homework Statement Solve inital value problem: x''+2x'+x=g(t) g(t)= t 0<t<1 2-t 1<t<2 0 t>2Homework Equations Second shift theorem, Heaviside function and Laplace transforms. I denote Heaviside,functuon H(t-a), and Laplace transform with L The Attempt at a Solution I...
  24. S

    A very basic laplace transform problem (picture included)

    Hi..new here. I'm having trouble finishing off a very basic laplace transform problem. I'd greatly appreciate it if someone could show me the way. The problem is #5 from this picture: http://www.flickr.com/photos/67208468@N08/6111476312/ What I did was use the Euler's formula for sin t and I...
  25. J

    Laplace Transform: Learn Eberhard Zeidler's Definition

    I am reading the book QFT II: Quantum Electrodynamics by Eberhard Zeidler. On page 32 he defines the Laplace transform as: F(s) = \int_0^{\infty}f(t)e^{ist}dt Where f is a smooth function and F is it's Laplace transform. I have changed his notation, but not the content of the formula. I...
  26. P

    Understanding Complex Func., Laplace Transforms & Cauchy Riemann

    I am reading a chapter on Complex Functions, Laplace Transforms & Cauchy Riemann (as part of Control theory) And I don't understand how they arrive at a particular part. [ I tried to type it out in tex, but it takes way too much time so uploaded a screenshot to flickr]...
  27. R

    Use the second shifting theorem to find the Laplace transform

    Homework Statement Use the second shifting theorem to find the Laplace transform of f(t) = {t2, t<4 {t, t\geq4 Homework Equations L{f(t - a)u(t - a)} = e-as F(s) The Attempt at a Solution Okay so I applied the unit step function to get the equation into the form f(t)...
  28. J

    Assumed solution for the Laplace EQUATION?

    Assumed solution for the Laplace EQUATION?? The book I'm using says that the method of separation of variables is a must when solving the Laplace equation. OK, well they ASSUME that the solution looks like V(x,y,z) = X(x)*Y(y)*Z(z) but why can't they assume a solution of V(x,y,z) =...
  29. R

    Solving an Initial value problem using Laplace transform

    Homework Statement Solve the Initial value problem using Laplace transform \ddot{y} +2y = 0, y0 = C1, \dot{y} = C2 Homework Equations [s2 - sy(0) - y'(0)] + a[sY - y(0)] + bY The Attempt at a Solution s2Y - sy(0) - y'(0) + 2y = 0 s2Y + 2Y = sy(0) + y'(0) (s2 + 2)Y = s(C1) + (C2)...
  30. D

    Differential equations with laplace

    Homework Statement D^2 x -2x = 2sin2t x(0) = 0 , x'(0) = 1 Homework Equations D^2 x = s^2ƒ[x] - sx(0) - x'(o) The Attempt at a Solution s^2ƒ[x] - sx(0) - x'(o) - 2ƒ[x] = 2sin2t s^2ƒ[x] - sx(0) - x'(o) - 2ƒ[x] = 4 / s^2 + 4 ƒ[x] (s^2 + 1) + 1 = 4 / s^2 +4...
  31. R

    Finding the inverse Laplace transform

    Homework Statement Find the inverse Laplace transform of the following function 1/[s(s2 + 4)2] Homework Equations 1/ (s2 + \omega2)2 = (1/ 2\omega3) (sin\omegat -\omegat cos\omegat) The Attempt at a Solution L-1 = (1/s) (1/ s2 +22)2 = (1/16) (sin 2t - 2t cos 2t) as the Laplace...
  32. Q

    Question about laplace transform limits

    if say the function f(t) is 0 from 1 to 2 is 1 from 2 to 3 if i laplace transform it , for f(t) = 1, i get f(s) = 1/s so what are the limits for my s ? is it still 2 to 3? thanks!
  33. M

    Finding the Inverse Laplace Transform (Bromwich Contour)

    Homework Statement Find the inverse laplace transform: {\cal L}^{-1}\{e^{-\sqrt{s-a}c}\} where a and c are constants. s is the complex variable.Homework Equations Bromwich integral: \frac{1}{2\pi{}i}\lim_{T\rightarrow \infty}\int_{\gamma-Ti}^ {\gamma+Ti}e^{st-\sqrt{s-a}c}ds...
  34. FeDeX_LaTeX

    The Laplace Transform of tan(t): Is Integration by Parts the Only Way?

    EDIT: I think all errors are fixed now. It's 1:30 AM and I'm going to bed now. Hello, I am trying to find the Laplace transform of tan(t), but I don't know if I'm getting anywhere. I can't find it in Laplace transform tables and WolframAlpha gives me an answer in terms of complex numbers...
  35. I

    How to Understand Laplace Transform Maths

    Just a quick question...what kind of maths do i need to understand to start to get to grips with laplace tranforms? Thanks
  36. N

    Laplace transform and HP50g

    Homework Statement Hi, I am writing a little program (I need it for my course from Theory of Reliability) and as I am neither experienced mathematician nor programmer I faced with this problem. What would this formula Look like if I would sole it for 'z' not for 'Fz' ? Homework...
  37. V

    Limit properties laplace transform

    hello, I'am stuying the laplace transform and have problems with understanding the follow thing my course says: lim s->+inf F(s)=lim t->0 f(t) lim s->0 F(s)=lim t->+inf. f(t) I would like to understand it because I have the feeling that this are important properties. Thank you very...
  38. C

    Inverse Laplace Transform

    I know that there is the Bromwich integral which inverts the Laplace Transform but it requires the function to have poles. I am wondering if there is a formula for inverting a function which decays rapidly but has no poles?
  39. R

    Laplace Transforms: Solving exp(-as)*(1/s+5) w/ S-Shifting

    To work out the laplace transform of exp(-as)*(1/s+5) Does convolution have to be used or can s - shifting be used. If s-shifting can be used can you please provide an example of how to use it in this case? My first though was s-shifting but then I would get exp(-as)*(1/[(s-a)+5])...
  40. A

    Solving the problemWhat is the Laplace Transform of t.H(t-a)?

    Homework Statement Find the Laplace Transform of t.H(t-a) where H is the heavyside (unit step) function. Homework Equations Properties of Laplace Transforms L{t.f(t)} = -Y'(s) L{f(t-a).H(t-a)} = e-as.F(s) Maybe another one I don't know about? The Attempt at a Solution...
  41. S

    Laplace Transforms for improper integrals?

    Homework Statement I (came up with)/(heard about) a way of using Laplace transforms that I didn't think about before. The problem is that it doesn't work for some reason. Look at following integral: I = \int_{0}^{\infty }sin(t)dt Say that you had no idea how to integrate something...
  42. D

    Laplace in Polar Coordinates: Solving the Polar Coordinate Equation

    Homework Statement solve the polar coord equation Homework Equations Urr+(1/r)Ur+(1/r2)U\vartheta\vartheta=0; 1\leqr\leq2, 0\leq\vartheta\leq\pi/4 U(1,\vartheta)=0 U(2,\vartheta)=\vartheta U\vartheta(r,0)=0 U(\vartheta,\pi/4)=0 and has the form...
  43. D

    Laplace equation involving a disk

    Homework Statement disk has a radius C with boundary conditions V(C,\vartheta)={cos(\vartheta) ;-\pi/2\leq\vartheta\leq\pi/2. 0; otherwise solve the laplace equation Homework Equations co/2+do/2ln(r)+\sum(cn1rn+dn1r-ncos(n\vartheta)+(cn2rn+dn2r-nsin(n\vartheta) The Attempt at a...
  44. A

    Inverting Laplace Transform: Y to f - Explained Steps

    Ok, so I have uploaded/attached the question and the solution. I just need help understanding the solution please. I understand how to calculate the initial inverse transform, but I included it as the reference to the second part of the question regarding the y'' + 4y' = H(t-3) Can someone...
  45. T

    Laplace Transform of e-t sin t: How to Derive the Denominator?

    Homework Statement Laplace Transform of e-t sin t Homework Equations The Attempt at a Solution I have the solution, but I am unable to figure out how the denominator becomes 1/[(s + 1)2 + 1]
  46. P

    Problem with finding an Inverse Laplace Transform

    I've been messing around with Laplace transforms. Anyway to get to the point I arrived at a "solution" in the s domain and got stuck. I'm trying to solve for the inverse laplace transform of A: {\cal L}^{-1} \{A\} where A = F(s) e^{C_2\sqrt{-s+C_1 }} and C_1,C_2 are constants and...
  47. Y

    Verify Laplace equation in rectangular coordinates.

    Just want to verify Laplace equation in rectangular coordinates that: \nabla ^2 \vec E = 0 \Rightarrow\; \nabla^2 \vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} +\frac {\partial^2}{\partial z^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) = 0 \hbox...
  48. B

    Laplace transform in electrical engineering

    Hello, We recently learned in mathematics the Laplace transform. But we didn't actually use it to solve circuits yet. I took things at my own hand and tried to figure out what's the point of Laplace transform. I figured out all that frequency and time stuff. But here is the thing. If...
  49. I

    Laplace transform proof of L{f'(t)}

    i can do the majority of laplace questions that will be asked, but i don't understand how to derive the formulae, and this is a past paper question :/ could someone please walk me through this question, or give me some tips?
  50. K

    C/C++ [Ask] How to write laplace transform in c or c++ ?

    [Ask] How to write laplace transform in c or c++ ?? Hi Guys.. if I have a laplace transform example : T(s) = 1/s+1. It's easy to solve or write in Matlab, but how to write it in C programm ?? thanks :)
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