The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

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Simple way to prove that $e^{-x^2}$ doen't admit Laplace inverse trasform

The question is already contained in the title. Is there any criterion that one can use to show this or is it necessary to apply Mellin's inverse formula and verify that the integral doesn't converge?
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Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
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Laplace transform of the square root of a generic function

Let $f(t)$ be a function (for example of time $t$). Is there a general expression of the laplace transform of $\sqrt{f(t)}$ ? Same question for the inverse Laplace transform : Let $f(s)$ be the ...
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Another Laplace transform of a function with square roots.

This question is very much related to this (one). Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{4+3s+\sqrt{s(4+s)}}.$$ My question is what is the inverse Laplace transform ...
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Laplace transform of $\cos(at)$

I need to find the Laplace transform of $\cos(at)$ I know that $L\{\cos(at)\}= \int_{0}^{\infty} e^{-st} \cos (at) dt$ but I am having trouble finding the integral Thank you
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Complicated Laplace Transform

I have found the following Laplace Transform in a list $$\int\limits_0^{\infty}e^{-st}\frac{e^{-u^2/4t}}{\sqrt{\pi t}}dt = \frac{e^{-u\sqrt{s}}}{\sqrt{s}}.$$ I am wondering how to prove this? I ...
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Laplace transform of a product of K simple functions

Is there a closed form expression for the Laplace Transform of a function which is a product of K simple functions, where the i-th function is of the form (1 - exp(-k_i *t))?
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Laplace Transform of $tf(t)$

Q. prove that $\mathfrak{L}\{tf(x)\}=-\frac{d\mathfrak{L}\{f(x)\}}{ds}$ where the notation used is standard one. Attempt I tried what would seem obvious way to start: $$\mathfrak{L}\{tf(x)\}=\int_0^\...
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Does the following have a Laplace transform?

I've looked at several resources and used Wolfram alpha but have been unable to find a Laplace transform for the following function: $$f(s) = {s\over \sqrt{a^2-\left({s\over 2}\right)^2}}$$ For a = ...
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29 views

Inverse laplace transform of $\dfrac{\alpha s}{s+\beta}$

I want to know the inverse laplace transform of $$\dfrac{\alpha s}{s+\beta}$$ where $\alpha, \beta$ are non-zero constants I already know the result for $$\dfrac{\alpha }{s+\beta}$$ Which is $\...
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40 views

Inverse Laplace transform of function with square roots.

Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{2+s+\sqrt{4s+s^2}}.$$ My question is, what is the inverse Laplace transform of $F$? From solving similar problems I have a ...
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How to calculate the Laplace transform

$$ h: t \in[0,+\infty[ \to \int_{t}^\infty \frac{1}{e^s\sqrt{s}}ds$$ I have to calculate the Laplace transform of $h$ in $0$ I know that $L[\int_{0}^\infty f(t)dt](p)= \frac{1}{p}L[f](p)$ but i ...
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How to derive through a convolution?

Let $f(t) = \alpha e^{-\beta t}$, where $\alpha, \beta$ are constants Let $g(t) = y(t)$ Then the resulting convolution $f\ast g$ is: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
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55 views

What is the inverse Laplace transform of $\lfloor s \rfloor$?

How can we find the inverse Laplace transform of: $[x]$ (floor function) ? My question isn't LLaplace transform of floor function i asked the "inverse" laplace transform of floor function $$\mathcal{...
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44 views

Laplace Transform Derivation Help

Please see image. These are screenshots of a lecture slide from a Control Engineering module, regarding determining the transfer functions of mechanical systems. However I can't seem to understand how ...
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264 views

Laplace transform nonlinear equation

How can I apply the Laplace transform on a the following nonlinear PDE $$ \frac{\partial y}{\partial t}=\frac{\partial y^n}{\partial x}$$ where $n$ is a natural number? When I say apply the Laplace ...
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Laplace transform of the Bessel function of the first kind

I want to show that $$ \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx = \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}}\ , \quad \ (n \in \mathbb{Z}_{\ge 0} \, , \text{Re}(a) >0 , \, b >0 ...
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Derivative property of Bilateral Laplace Transform

According to the definition of the bilateral Laplace Transform: $$ X(s)=\int_{-\infty}^{+\infty}x(t)e^{-st}dt$$ where $s=\sigma+i\omega$. So to get the derivative property, $y(t)=\frac{dx(t)}{dt}$: $$...
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Solving ode with Laplace transform

What i want to ask is both question 20 and 21 with Laplace transforms Actually i can solve question 20 and 21 with method of undetermined coefficients but . In laplace transform, i don't know how to ...
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1answer
26 views

Inverse Laplace Transform with squared irreducible quadratic in denominator using convolution theorem

Please help me find the inverse Laplace transform of $$F(s) = \dfrac{4}{(s^2+2s+5)^2}.$$ The answer I got is $\frac 1 5 (e^t - e^{-t}) \cos 2t - \frac 1 2 (e^t + e^{-t}) \sin 2t$. I first applied ...
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Laplace Transform with initial value

Use the Laplace transform to solve the following initial value problem: $$y'' + y = 2t$$ with $y(\pi/4) = \pi / 2 $ and $y'(\pi/4) = 2 - \sqrt{2}$. I understand this type of problems ...
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Solve a Laplace in Poolar Coordinate under two minutes?

I trouble with calculating the following example from previous exam with short solution on this Link. OP says there is a Laplace ٍPoolar Coordinate: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\...
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Find the leading order asymptotic behaviour of the integral

$$I(x) = \int_0^{\infty}e^{-t-\frac{x}{t^2}}dt \mbox{ as } x \mbox{ tends to infinity} $$ I know this has a moveable maximum so you need to make a substitution which transforms it into the integral: $$...
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Does the scaling property of Laplace transforms also hold for complex scaling?

Consider the following Laplace transform (it arose in the context of Borel resummation) $$ \int_0^{\infty}e^{-\zeta}\phi(z\zeta)d\zeta $$ my textbook says that $$ \int_0^{\infty}e^{-\zeta}\phi(z\zeta)...
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Initial Value Problem using Laplace transformation: What is the ${\cal L}$ transform of $u(t-5)$?

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses a single difficulty. If it's any indication of difficulty, the exercise ...
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64 views

Converse of the Watson's lemma

Watson's lemma basically says $$ f(t) \sim t^{\alpha} \,\,\,(\text{for small } t) \implies \int_0^{\infty} f(t) e^{-st} dt \sim \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}} \,\,\,(\text{for large } s). $...
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Asymptotic behavior of inverse laplace transform [duplicate]

My question may be quite rough. Let $F(\lambda)$ be the Laplace transform of some function $f(t)$, $$ F(\lambda)= \int_0^\infty e^{-\lambda t}f(t) dt. $$ If I have knowledge about $F(\lambda)=O(\...
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Statistics - Laplacian Characteristic Function

I was asked to find the characteristic function of the Laplacian random variable. And, from that, to find the mean of it. Im having trouble solving this. That's what I have done so far, but it doesn'...
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Inverse Laplace transform of $(s^2-1)^{-1/2}$

please help with this. Not derived from any differential equation. Also found the answer $\mathcal{L}^{-1}(\dfrac{1}{\sqrt{s^2-1}})$
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How to calculate the inverse Laplace transform of $F(s)=\frac{1}{1-e^{-s}}$?

How to calculate the inverse Laplace transform of this function? $$F(s)=\frac{1}{1-e^{-s}}$$
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Trigonometric functions and complex numbers

I solving the inverse Laplace transform using the method of Heaviside. This is part of the problem: I understand the division between complex numbers and that $e^{it} = Cos(t) + iSin(t)$, but I ...
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How to solve a differential equation with a distributional free term?

I tried to solve this type of differential equation $$y'' + y = \delta + \delta' .$$ I tried using the Laplace Transform, but I'm stuck at that $\delta$ (Dirac function). The only thing I know is ...
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What exactly is the variable of Laplace transform

I try to find the solution for a hard differential equation. I could not solve it with any orthodox method. However, if I use Laplace transform and then replace its term with its Maclaurin series, it ...
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Laplace and Transfer Function Problem

I have to show that all initial condition problem where $t = 0$ with constant coefficients: $ a_{n} y^{n}(t) + a_{n-1}y^{n-1}(t)+...+a_{2}y''(t) + a_{1}y'(t) + a_{0}y(t) = x(t) $ with $ y^{k} = y_{...
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extracting a function out of an equation

I encountered the following problem in my thesis. We have an equation as follows: $\phi(s)=\sum^\infty_{n=1}P(n)\int^{\infty}_{0}e^{-st}f(t|n)dt=\sum^{\infty}_{n=1}[(1-q)M_1(s)]^{n-1}qM_2(s)$ in ...
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Solving convolution problem with $\delta(x)$ function

Suppose we had the functions: $$g(t)=\theta(t)(e^{-t}+2e^{-2t})+2\delta(t)$$ and $$u(t)=2(\theta(t)-\theta(t-2))$$ Then we have $$u*g=\int_{-\infty}^{\infty}g(\tau)u(t-\tau)d\tau=2\int_{t-2}^{t}(e^{-\...
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Solve non-linear pde

so i wonder the next thing : if i consider a pde like heat equation Fourier transform works very well. Now, if i consider this equation : $\frac{\partial u(t,x)}{\partial t}-k(u(t,x))\frac{\partial^2 ...
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Laplace transform of floor$(x)$

One way to compute the Laplace transform of floor$(x) = \left \lfloor{x}\right \rfloor$ (defined as the greatest integer $\leq x$) for $x$ positive is to use the definition of the transform. Write ...
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How do I proove that function is solution of the Laplace equation?

How do I proove that for $\vec{r}=(x,y,z)\in \mathbb{R}^3,\vec{r}\neq 0$, function is $u(x,y,z):=1/(-ln\left \| \vec{r} \right \|)$ a solution of the Laplace equation $\Delta u=\frac{\partial^2 u}{\...
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Laplace Transform of $t^2$ , for $t\ge1$ .

$$f(t) = t^2 , t>=1$$ $$f(t) = 0, 0<t<1$$ what is the laplace transform of $f(t)$, It is solved In my sheet as $$t^2 = (t-1)^2 + 2t -1 $$ $$ L(f(t)) = e^{-s} L(t^2 +2t +1) = e^{-s}(2/s^3 + ...
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Laplace transform to solve a pde

i have to find the laplace transform of this : $$f(u(x,t))\frac{\partial u(x,t)}{\partial t}$$ I have no idea how to solve this... Thank you for your help.
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Laplace transform using change of scale property [closed]

If Laplace transform of $f(t)=\phi(s)$, then Laplace transform of $e^{bt}f(at)$ is
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prove the result of a Laplace transformation

I have to prove the next problem $$\mathcal{L} \left(\int_{0}^{t}\frac{1-e^{-u}}{u}du,s\right) = \frac{1}{s}\log\left(1+\frac{1}{s}\right)$$ I'm quite new in the subject and I have troubles with ...
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Laplace Transform on this Poisson Process?

Two independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$. The waiting is exponentially distributed with mean of $\frac{1}{\mu_1}$ and $\frac{1}{\mu_2}$. Knowing this, how to ...
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Why is Laplace Transform of $\delta(t)$ $F(s)=1$, not $0$ or $\frac{1}{2}$?

Let $\varepsilon \in \mathbb{R}$, the following integral $$ \int_{\varepsilon}^\infty \delta(t)e^{-st}dt. $$ converges to $1$ if $\varepsilon \to -0$ and $0$ if $\varepsilon \to +0$. This shows ...
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laplace transform probability distribution not concentrated on 0

This seems intuitively obvious but how to prove that $\hat{\mu} < 1,$ when $\theta >0$ and $\mu$ is a probability measure not concentrated at $0,$ where $\hat{\mu}$ is defined as below $$\hat{\...
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Get transfer function of a nonlinear diff. equation

I have this equation: $$\frac{\partial v}{\partial t} = -g + c\left(u(t) - v(t)\right)^2$$ g and c are constants. u(t) is my input and v(t) is my output. I need to reach the transfer function $\frac{...
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Laplace transform of $1/t$

Does the laplace transform of $1/t$ exist? If yes, how do we calculate it? Putting it in $$\int_0^\infty (e^{-st}/t) dt$$ won't solve. Is there any other way? If not, why? Thanks!
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Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...