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. J

    Inverse Laplace Transform of F(ks)

    Homework Statement Compute the inverse laplace transform of F(ks)Homework Equations The Attempt at a Solution $$L^{-1} (F(ks)) = \frac{f}{|k|} \left( \frac{t}{k} \right)$$ Correct?
  2. J

    Connection between unilateral laplace

    Exist some conection between: $$\int_{0}^{+\infty} f(t) \exp(-st)dt\;\;(1)$$ $$\int_{-\infty}^{0} f(t) \exp(-st)dt\;\;(2)$$ ? The results, the transformations, are very similar, with some little difference in the signal. So, known the transformation (1), is possible to find the (2)?
  3. J

    Why inverse laplace is line integral?

    Watching this video http://youtu.be/1JnayXHhjlg?t=5m30s, I understood the ideia the Fourier transform, that is a continuous summation of sinusoids. But now If I have amplitude and phase as function of σ and ω, the summation wouldn't be ##\sum_\sigma \sum_\omega A_{\sigma \omega} \exp(i...
  4. baby_1

    Laplace in electromagnetics(voltages are different in conductor?)

    Hello as you see this example and solution if we assume the R of conductor sphere is 5m and check voltage in different z we obtain(for example z=1m and z=2) and as we know in conductor we doesn't have voltage differences so this equation should be the same C=0 but voltage...
  5. 1

    Unsure about Inverse Laplace Heaviside Function question

    Homework Statement Find the inverse Laplace transform of F(s)=5e^(-8s)/(s2+36) Homework Equations The Attempt at a Solution I know that to find the inverse Laplace transform of this function, I start by factoring out (e^(-8s)) to end up with 5/(s^2+36), and that my final answer...
  6. S

    Solving ODEs Using Laplace: Two Challenging Problems

    I have the following 2 problems to solve using Laplace. 1) x'' + 3x' +2x=e^(-t); with x=dx/dt=0 when t = 0 2) x'' - 2x' +10x=e^(2t); with x=0 and dx/dt=1 when t=0 Where x' is dx/dt and x'' is the second derivative against time. My attempts: 1)Using laplace I get...
  7. M

    What Is the Intuitive Explanation of the Laplace Transform?

    hello pf! i am wondering if anyone here knows of a geometric, intuitive explanation for the laplace transform? if so, please direct me to the source of if you could, explain to me your understanding? thanks!
  8. J

    Electromagnetism: find the laplace force

    Hi every one. Homework Statement in french,the problem says:"déterminer la force de laplace qui s'exerce entre deux fils de courants réctilignes,indéfinis,parraléles d'intensités I et I' de meme sens".maybe in english it means:"find the laplace force that is applied between two...
  9. P

    MHB Douglas' questions about Laplace Transforms

    1. $\displaystyle \begin{align*} y''(t) + 7y'(t) + 12y(t) &= -36H(t - 2) \textrm{ with } y(0) = 0, y'(0) = 0 \\ \mathcal{L} \left\{ y''(t) + 7y'(t) + 12y(t) \right\} &= \mathcal{L} \left\{ -36H(t-2) \right\} \\ s^2\,Y(s) + s\,y(0) + y'(0) + 7 \left[ s\,Y(s) - y(0) \right] + 12Y(s) &=...
  10. 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(...
  11. vanceEE

    Derivative of Laplace Transforms

    Homework Statement $$xy'' + xy = 0$$Homework Equations y'(0) = 0, y(0) = 1The Attempt at a Solution $$ L[xy''] + L[xy] = 0$$ $$-L[-xy''] - L[-xy] = 0$$ $$-\frac{d}{dp}L[y''] - \frac{dY}{dp} = 0 $$ $$-\frac{d}{dp}(-y'(0)-py(0) + p^2Y) - \frac{dY}{dp} = 0 $$ $$\frac{d}{dp}(-p + p^2Y) +...
  12. P

    MHB Douglas' question regarding Laplace Transforms (1)

    Hi Douglas, I agree, partial fractions would be the best approach. Notice that $\displaystyle \begin{align*} s^4 + 1024 &= \left( s^2 \right) ^2 + 2\cdot s^2 \cdot 32 + \left( 32 \right) ^2 - 2\cdot s^2 \cdot 32 ^2 \\ &= \left( s^2 + 32 \right) ^2 - 64s^2 \\ &= \left( s^2 + 32 \right) ^2 -...
  13. vanceEE

    Find Limits of Integration in Polar Coordinates with Laplace Math Help

    px = t t = s^2 $$ I = \int_0^∞ e^{-s^2}ds$$ $$I*I = \int_0^∞ e^{-s^2}ds * \int_0^∞ e^{-u^2}du = \int_0^∞\int_0^∞ e^{-(s^2+u^2)}du ds$$ $$s = rsin\theta $$ $$u = rcos\theta $$ $$r = s^2 + u^2 $$ $$ I*I = \int_0^∞\int_\alpha^\beta e^-{r^2}rdrd\theta$$ How can I find my limits of integration in...
  14. D

    MHB Region of Convergence and Inverse Laplace

    How do I find all the possible ROC for a transfer function written as \[ H(s) = \frac{(s - 2)^{n_1}}{(s + 2)^{n_2}(s + 1)^{n_3}(s - 1)^{n_4}} \] where \(n_i\in\mathbb{N}\).
  15. D

    MHB Solving Laplace Transform: \[(s+1)^2/(s^2-s+1)\]

    \[ \frac{(s+1)^2}{s^2 - s + 1} \] I have simplified it down to \[ \frac{s - \frac{1}{2} + s^2 + s + \frac{3}{2}}{(s - 1/2)^2 + \frac{3}{4}} = e^{1/2t}\cos\Big(t\frac{\sqrt{3}}{2}\Big) + \sqrt{3}e^{1/2t}\sin\Big(t\frac{\sqrt{3}}{2}\Big) + \frac{s^2 + s}{(s - 1/2)^2 + \frac{3}{4}} \] but I can't...
  16. 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 \[...
  17. R

    Insulated boundary for circular laplace equation?

    Homework Statement Consider the Laplace’s equation, ∆u(r,θ) = 0, inside the quarter-circle of radius 2 (0 ≤ θ < π, 0 ≤ r ≤ 2), where the boundary θ is insulated, and u(r,\theta/2)=0 Show that the insulated boundary condition can mathematically be expressed as \frac{\partial u}{\partial...
  18. A

    Laplace transform initial value problem-

    Laplace transform initial value problem--need help! Looking at the solutions to these initial value problems, I am very confused as to how the highlighted steps are derived (both use heaviside step functions). I know the goal is to get the fractions in a familiar form so that one can look them...
  19. polygamma

    MHB Evaluating an Integral: Laplace Transform Method

    The typical way to evaluate $ \displaystyle \int_{0}^{\infty} \frac{\cos mx}{a^{2}+x^{2}} \ dx$ is by contour integration. In a recent thread I evaluated that integral using the Laplace transform. http://mathhelpboards.com/analysis-50/advanced-integration-problem-9129.html#post42551My...
  20. T

    Engineering How Can Laplace Transforms Be Used to Analyze Circuit Transient Responses?

    The amplifier and filter of FIGURE 4 can be represented by the equivalent circuit of FIGURE 6. (Figs are in image 1 and 2 attached) (i) Draw the Laplace equivalent circuit and hence derive the equation for vout(s) in terms of R, L and C for a step input. (ii) Attempt, using the component...
  21. T

    How Does the Hankel Transform Solve Problems in Cylindrical Domains?

    Hello, As of recently, I've been working with Laplace transforms and have a question about their relationship to solving differential equations. I know the definition of the laplace transform and I know that a function is essentially being transformed from the time domain to complex...
  22. T

    Arranging this expression into a laplace transform

    Hi Guys, I have an expression that i am struggling to manipulate into a laplace transform. This expression should fit one or a combination of the common transform pairs. I believe the transform the expression should be fitting is either a unit step 1/s a unit ramp 1/s^2 an exponential 1/s+a...
  23. D

    Derivation of Laplace in spherical co-ordinates

    I have been trying to derive the Laplace in spherical co ordinates. I have attached a file which has basic equations. I am trying to get the following equation. d(phi)/dx= -sin(phi)/(r sin (theta)). I have also attached the materials I am referring to. Can someone please help me derive...
  24. B

    Any suggestions for finding the inverse Laplace transform of 11/(s^2+16)^2?

    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?
  25. B

    Inverse Laplace for (e)^-5t*(t)^4

    Homework Statement Find: Inverse Laplace for x(t)= (e)^-5t*(t)^4 using laplace table and laplace properties. Homework Equations The Attempt at a Solution Well, I have been working on this problem for a few days now and cannot seem to figure it out. The two functions are not...
  26. B

    Laplace transform with 2s(s^2+9) in denominator

    Homework Statement Hi I would like to know how to expand 20/2s(s^2+9) in order to find the inverse Laplace Transform of the function K(s) to gt k(t). The (s^2+9) term in the denominator is throwing my calculations off for the constants because of the s^2 term. Homework Equations...
  27. B

    Laplace Transform for shifted Unit Step Function

    Hello, I have a relatively simple question. after being unable to find it through google I have decided to ask you guys if you know what the Laplace transform of a unit step function that looks like this would look like Us(t-2) From tables, the Laplace transform for a regular units step...
  28. 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
  29. D

    MHB Laplace Transform with two steps

    Given the transfer function \[ H(s) = \frac{Y(s)}{U(s)} = \frac{1}{s + 1} \] and \[ u(t) = 1(t) + 1(t - 1). \] How do I find U(s)? I know I take the Laplace transform of u(t) but with the two step functions how can this be done? The Laplace transform of the step function is \(\frac{1}{s}\)...
  30. D

    MHB Inverse Laplace Transform problem

    I can't seem to part of an inverse Laplace transform correct. \begin{align*} f(t) &= \frac{6}{5}\mathcal{L}^{-1}\bigg\{\frac{1}{s + 2}\bigg\} + \frac{3}{5}\mathcal{L}^{-1}\bigg\{\frac{3s - 1}...
  31. G

    Laplace Transform Within a Domain

    Homework Statement Find the Laplace transform of f(t) = t \forall 0≤t≤T, 0 otherwise Homework Equations The Attempt at a Solution I write the function as tu(t)-t*u(t-T) That is turn on the function t at t=0 and turn the function t off at t=T. It seems to be right to me...
  32. D

    Solve a Laplace transform puzzle

    Hi~ I recently solve a Laplace transform problem as following L[int{t,0}cosh(t'-1)U(t'-1)dt']=? U(t'-1) is the unit step function(=1 for t'>1, =zero otherwise) According the standard Laplace formula : (1)L[cosh(t-1)U(t-1)]=exp(-s)*s/(s^2-1); (2)L(int{t,0}f(t')dt')=F(s)/s...
  33. U

    Laplace Equation, potential around cylinder

    Homework Statement Part(a): State condition for laplace's to work. Find potential in space between electrodes. Part (b): Find potential inside cylinder and outside. Part (c): How would the answer change is one pair is kept at potential 0 while other V0? Homework Equations The Attempt at a...
  34. I

    Trouble with Laplace transform of a function

    I have a probability distribution as follows: \begin{equation}p_j(t)=\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{1,k}e^{ij\tilde{k}}\right|^2+\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{2,k}(t)e^{ij\tilde{k}}\right|^2\end{equation} where...
  35. alyafey22

    MHB Existence of Laplace transform

    Prove the following Suppose that $f$ is piecewise continuous on [0,\infty) and of exponential order $c$ then \int^\infty_0 e^{-st} f(t)\, dt is analytic in the right half-plane for \mathrm{Re}(s)>c
  36. P

    Solving Integral Using Laplace Transform

    I posted this in the homework section, but I haven't received any help, so hopefully putting it in this section won't be an issue. I'm trying to compute the integral $$ \int_0^ \infty \frac{ \cos xt}{1 + t^2} \, dt $$ using the Laplace transform. The first thing that catches my eye is the 1...
  37. P

    What is the Laplace Transform Integral for the function $\frac{\cos xt}{1+t^2}$?

    Homework Statement $$ \int_0^\infty \frac{\sin xt}{x} \, dt $$ Homework Equations The Attempt at a Solution $$ = \int_0^\infty L(\sin xt) \, dp $$ $$ = \int_0^\infty \frac{x}{p^2 + x^2} \, dp $$ $$ = x \int_0^\infty \frac{dx}{p^2 + x^2} \, dp $$ p = x tan theta: $$ = x \int_0^{\pi/2}...
  38. P

    Question: How do we use the Laplace transform to find the inverse of a function?

    Homework Statement Find $$ L^{-1} \left[ \frac{1}{ (p^2 + a^2)^2} \right] $$ Homework Equations $$ L [ x \cos ax ] = \frac{p^2 - a^2} { (p^2 + a^2) ^2 } $$ The Attempt at a Solution I have no idea. Any thoughts?
  39. J

    Correcting a Laplace Transform Problem

    Homework Statement Why I am getting wrong answer related to this Laplace Transforms problem? According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get (e^{-st}/(s^{2} + a^{2}))(a sin at - s cos at)Homework Equations ∫e^{-st}cos at dt The...
  40. Y

    Laplace and Poisson's equation.

    Homework Statement I want to verify for v=\ln( r), that a)\nabla^2 v=0 for v on a disk center at (x_0,y_0). Therefore v is Harmonic. b)\nabla^2 v=\frac{1}{r^2} for v on a sphere center at (x_0,y_0,z_0). Therefore v is not Harmonic.The Attempt at a Solution a) For circular disk...
  41. M

    Laplace Differential Equation of a Half-Annulus

    Here is the DE: Δu(r,θ)=0, 1 ≤ r ≤ 2, 0 ≤ θ ≤ pi and here are the Boundary Conditions: u(1,θ)=sin(θ), u(2,θ)=0, u(r,0)=0, u(r,pi)=0 Based on the Boundary Conditions I believe this is half of an annulus. Using the 2D Laplace equation for polar coordinates, find the solution u(r,θ). I've...
  42. I

    Laplace Transform IVP (Easy I think)

    Homework Statement Use Laplace transforms to solve the initial value problems. ##y''+4y=0;## ##y(0)=5;## ##y'(0)=0## Homework Equations The Attempt at a Solution $$y''+4y=0$$ $$L(y'')+L(4y)=L(0)\implies s^{2}Y(s)-sy(0)-y'(0)+4(sY(s)-y(0))=0\implies s^{2}Y(s)+4sY(s)-5s-20=0$$...
  43. T

    Laplace Transform Solution to Second Order ODE IVP

    Homework Statement y''+6y=f(t), y(0)=0, y'(0)=-2 f(t)= t for 0≤t<1 and 0 for t≥1 Homework Equations The Attempt at a Solution L{y''}+6L{y}=L{t}-L{tμ(t-1)} where μ(t-1) is Unit Step Y(s)=L{y} sY(s)-y(0)=L{y'} and y(0)=0 s2Y(s)-sy(0)-y'(0)+6Y(s) where y(0)=0 and...
  44. 1

    Inverse Laplace Transform of s/(s^2+1)^2)

    Homework Statement ##\mathcal{L}^{-1}\Big\{\frac{s}{(s^2+1)^2}\Big\}## I'm trying to figure out how to find the inverse Laplace transform of this expression. Is this something you just look up in a table or is there a way to find it directly, maybe by Convolution?
  45. L

    Laplace transform interpretation

    Hello, We were introduced to Laplace transforms in my ODE class a few days ago, so naturally I went online to try to figure out what this transform actually is, rather than being satisfied with being able to compute simple Laplace transforms. The Wikipedia page for the transform says it...
  46. I

    Laplace Transform of t: Using Integration by Parts

    Homework Statement Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph). 1) f(t)=t Homework Equations The Attempt at a Solution Seems pretty easy... Question is, I don't understand the directions exactly.. Am I...
  47. Y

    Question on Laplace eq. in a ball

    For ##\nabla^2 u(r,\theta, \phi)=0##, ##u(r,\theta, \phi)=r^{n}Y_{nm}(\theta,\phi)##. But I have issue with this, for spherical coordinates: \nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial...
  48. G

    Laplace transform of Heaviside function

    Homework Statement What is the laplace transform of H(-t-17) Homework Equations Shifting theorem: L(H(t-a)) = (e^-as)/s The Attempt at a Solution This is the only part of the problem that I can not get (this part is from a larger differential equation I'm trying to solve). I'm can't seem...
  49. T

    Find the Inverse Laplace of 1/(s^3)

    Find the Inverse Laplace of 1/(s^3) is there some special rule for cube? The answer is t^2/2 Looking at the Laplace Table t^n looks similar but its not it exactly. What should I do?
  50. Rapier

    Laplace Transforms with an Undamped Spring with Unit Impulse

    Ok, so I start out with the basics and find k. F=ma=kx (1.5 kg)(9.8 m/s^2) = k (4.9 m) k = 3 N/m I also know from the problem that c=0 (no damping) and x(0) = 2m and x'(0) = 0. ƩF = ma ma = -cx' - kx, a = x'' mx'' + cx' + kx = 0 and since c=0 mx'' + kx = 0 Nothing out of the...
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