The Laplace transform is a widely used integral transform, similar to the Fourier transform.

learn more… | top users | synonyms

16
votes
4answers
8k views

What exactly is Laplace transform?

I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out ...
14
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
13
votes
1answer
938 views

Compute the inverse Laplace transform of $e^{-\sqrt{z}}$

I want to compute the inverse Laplace transform of a function $$ F(z) = e^{-\sqrt{z}}. $$ This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
10
votes
1answer
303 views

Laplace transform identity

Is there a function equal to its Laplace transform? I mean $$ \int_{0}^{\infty}dt\exp(-st)f(t)= f(s).$$ Of course I know $f(t)=0 $ satisfy the equation. For the case of the Fourier transform, I ...
10
votes
1answer
223 views

Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
9
votes
1answer
302 views

Is the Laplace transform a functor?

I may be oversimplifying, as I know very little about category theory, but: Does the Laplace transform, which—to my limited recollection—is a morphism between differential equations and algebraic ...
9
votes
0answers
1k views

Physical interpretation of Laplace transforms

One may define the derivative of $f$ at $x$ as $\lim\limits_{h\to0}\cdots\cdots\cdots$ etc., and show that that has certain properties, but it also has a "physical" interpretation: it is an ...
8
votes
5answers
848 views

How to find the Laplace transform of $\frac{1-\cos(t)}{t^2}$?

$$ f(t)=\frac{1-\cos(t)}{t^2} $$ $$ F(S)= ? $$
7
votes
1answer
59 views

Finding the inverse Laplace transform of $\frac{s^2-4s-4}{s^4+8s^2+16}$

$$F(s) = \frac{s^2-4s-4}{s^4+8s^2+16}$$ My work is as follows, $$\frac{s^2-4s-4}{(s^2+4)^2}=\frac{s^2+4}{(s^2+4)^2}-\frac{8}{(s^2+4)^2}-\frac{4s}{(s^2+4)^2}$$ The inverse laplace of the first term ...
7
votes
2answers
816 views

Bringing a limit inside of an integral

Under what conditions does $$ \lim_{a \to 0^{+}} \int_{0}^{\infty} f(x) e^{-ax} \ dx = \int_{0}^{\infty} f(x) \ dx \ ?$$ For example, for $a>0$, $$ \int_{0}^{\infty} J_{0}(x) e^{-ax} \ dx = ...
7
votes
1answer
235 views

Laplace transform of a random variable

My professor says that the Laplace transform of a nonnegative RV uniquely determines the RV up to distributional equality among all nonnegative RVs. He says one can argue this by appealing to a fact ...
7
votes
1answer
272 views

ODE Laplace Transform an impulse bring oscillating system to rest

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
6
votes
5answers
804 views

How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$?

How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
6
votes
1answer
360 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
6
votes
2answers
2k views

Why the Fourier and Laplace transforms of the Heaviside (unit) step function do not match?

The Fourier transform of the Heaviside step function $u(t)$ is $\dfrac{1}{iω} + π δ(ω)$. The Laplace transform of the same function is $\dfrac{1}{s}$. I remember the proof came from derivatives and ...
6
votes
2answers
246 views

Usage of inverse Laplace transform

At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof? ...
6
votes
2answers
242 views

Inverse Laplace Transform help

Is the information below correct? Find the inverse Laplace transform of $$ F(s) = \frac{s}{s^2 + 4s + 13}$$ Soln: a) Complete the squares to simplify our denominator $$ s^2 + 4s + 13 = (s+2)^2 + 9 ...
6
votes
1answer
111 views

Find the Laplace transform of $f(t) = \begin{cases} 0, & \text{if $t<5$} \\ t^2−10t+31, & \text{if $t\ge 5$} \\ \end{cases} $

Find the Laplace transform of $$f(t) = \begin{cases} 0, & \text{if $t<5$} \\ t^2−10t+31, & \text{if $t\ge 5$} \\ \end{cases} $$ $F(s)=$ __________? Here is my work. I went wrong ...
6
votes
1answer
267 views

solving the PDE of a beam under a moving load using Laplace transform

Solve this PDE using Laplace transform : $$ EI {\partial^4 y(x,t)\over\partial x^4}+\mu {\partial^2y(x,t)\over\partial t^2}= F(x,t) $$ $$F(x,t)= P\delta(x-u) / ...
6
votes
1answer
79 views

Find the inverse Laplace Transform of the following

Find the inverse Laplace Transform: $$\mathcal L^{-1} \left\lbrace 1\over s^4\right\rbrace$$ I used the equation: $$\mathcal L\left\lbrace t^n\right\rbrace={n!\over s^{n+1}}$$ and played with some ...
5
votes
3answers
242 views

Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
5
votes
2answers
507 views

Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $

I have a trouble with a integral: Using this Laplace trasform equation: $$\begin{align} \int_0^\infty F(u)g(u) \, du & = \int_0^\infty f(u)G(u) \, du \\[6pt] L[f(t)] & = F(s) \\[6pt] ...
5
votes
2answers
184 views

What kind of book would show where the inspiration for the Laplace transform came from?

I'm trying to find out where to learn about integral transforms and inversions like the Laplace transform and the Bromwich integral. I'm looking for a book that describes how you can find (derive) ...
5
votes
3answers
385 views

Does the Laplace transform biject?

Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.' Can someone provide a proof or counterexample ...
5
votes
2answers
499 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...
5
votes
1answer
1k views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
5
votes
3answers
489 views

Physical meaning behind Frequency domain?

I understand its usage and why is it important because It transforms differential equations to algebraic ones.. But I can't get the physical meaning of the new form of the equation and the meaning of ...
5
votes
2answers
98 views

Laplace Transform

How can one show that 1/$e^s$ is not the laplace transform of any function? Note that function here does not include distributions like dirac delta function.
5
votes
1answer
1k views

inverse Laplace Transform: $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$.

I am styding Laplace transforms and for some reason I have stuck in the followning exercise. Find the inverse Laplace Transform $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$. Any help? Thank's in ...
5
votes
2answers
479 views

Laplace transform. How to derive

I have this integral related to a Laplace transform and I was wondering if anyone knows of a clever way to derive it. I know we usually look these up in a table, but this form is not in a table I ...
5
votes
1answer
52 views

Initial Value Problem with Laplace Transform

How do you solve the following with Laplace Transform? $$ {\rm y}''\left(t\right) - 10\,{\rm y}'\left(t\right) + 25\,{\rm y}\left(t\right) = 24\,t\,{\rm e}^{-2t}\,; \qquad\qquad {\rm y}\left(0\right) ...
5
votes
1answer
298 views

Laplace transform and differentiation

Let $F(s)$ be the Laplace transform of $f(t)$: $$F\left(s\right)=\int_{0}^{\infty}e^{-st}f\left(t\right)dt$$ It then follows that $f(t)$ can be recovered from $F(s)$ by the inverse Laplace ...
5
votes
1answer
65 views

Laplace transformation $y''+2y'+2y=3\sin x+\cos x$

Given$$y''+2y'+2y=3\sin x+\cos x$$ Transform to image region $$Y(s)(s^2+2s+2)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ $$Y(s)((s^2+2s+1)+1)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ ...
5
votes
1answer
115 views

A Laplace transform question

Suppose I have a positive integrable random variable $X$ s.t. $$E[e^X]=+\infty$$ Now let's take a series with general term $p_n$, summing to one, and define $$Z=\sum_{n>0}p_ne^{X_n}$$ and $U=\ln Z$ ...
5
votes
2answers
358 views

Laplace transform of integrated geometric Brownian motion

Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ? A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where ...
5
votes
1answer
297 views

$\mathcal{B}^{-1}_{s\to x}\{e^{as^2+bs}\}$ and $\mathcal{L}^{-1}_{s\to x}\{e^{as^2+bs}\}$ , where $a\neq0$

http://en.wikipedia.org/wiki/Integral_transform#Table_of_transforms claims than the integral form of inverse bilateral Laplace transform and inverse Laplace transform are both the same. But are they ...
5
votes
1answer
140 views

Show that $\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} =\ln\frac{b}{a}$ [closed]

It should be using Laplace transform. I found similar problems already solved but I need this to be shown using Laplace transforms: $$\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} = \ln\frac{b}{a}$$
5
votes
0answers
272 views

Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...
4
votes
2answers
424 views

Find the inverse Laplace transformation of $\dfrac{s+1}{(s^2 + 1)(s^2 +4s+13)}$

My question is : find the function $f(t)$ that has the following Laplace transform $$ F(s) = \frac{s+1}{(s^2 + 1)(s^2 +4s+13)} $$ thanks
4
votes
1answer
205 views

Inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$ (Bromwich integral)

I am looking for the inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$. I started by using the general formula of the Bromwich integral: $\displaystyle \lim_{R\to\infty} \int_{a-iR}^{a+iR} ...
4
votes
1answer
182 views

Inverse Laplace transform computation

Calculate the inverse Laplace transform $$ \displaystyle{ \mathcal{L^{-1}} \{ s\log \frac{s^2 + a^2}{s^2 - a^2} \} }$$ I know that is boring but I would really appreciate some help. Thank's in ...
4
votes
1answer
2k views

Solving a differential equation with the Dirac-Delta function without Laplace transformations

So I'm trying to solve the following differential equation: $y''+3y'+2y=\delta(t-1)$, $y(0)=0$, $y'(0)=0$. (where $\delta$ is the Dirac's delta function) Everything I've read in my ...
4
votes
1answer
26 views

Deriving Laplace Transform of Laguerre polynomial

I'm given this definition for the Laguerre polynomials: $$L_n(t)=\frac{e^t}{n!}\frac{d^n}{dt^n}\left[t^ne^{-t}\right],~\text{for }n=0,1,2...$$ and I have to show that the Laplace transform is ...
4
votes
3answers
417 views

Integrating using Laplace Transforms

$$\int_{0}^\infty {\cos(xt)\over 1+t^2}dt $$ I'm supposed to solve this using Laplace Transformations. I've been trying this since this morning but I haven't figured it out. Any pointers to push me ...
4
votes
1answer
535 views

How to solve $t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$

I want to solve this equation. It reminds me something about Laplace transform. I am sure that I must use it order to solve it. $$t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$$ How to do it? ...
4
votes
2answers
387 views

Laplace Transformations

can someone kindly help me with these few questions? :) Find $L{e^tf(t)}$ in terms of f*(s) and state a range of s which this is defined. I couldn't figure this out. You use the definition but i ...
4
votes
1answer
59 views

Laplace transform of $f'(t)/t$

A question regarding the computation of $\mathcal{L}_s[f'(t)/t]$, where $f(t)$ is a differentiable function, was asked few hours ago. Unfortunately, this question was voluntarily deleted by the OP. I ...
4
votes
3answers
199 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
4
votes
2answers
130 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
4
votes
2answers
4k views

Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...