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

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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 ...
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Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
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Inverse Laplace Transform as Bromwich Integral

I am seeking a references that provide a rigorous treatment of the inverse Laplace transform (Bromwich integrals), and how to compute them (beyond using tabled solutions - they don't cover my needs, ...
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Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
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the continuity theorem with respect to Laplace transform

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{N}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit ...
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What are origins of the Laplace variable $s$

When I learnt the Laplace Transform I was just told the very standard formula that: $F(s) = \int_{-\infty}^{\infty}f(t) e^{-st} dt$. From this we went on to the table of transforms at its properties ...
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Inverse Laplace Transform of $e^{c \cdot s^2}$

I am trying to find the Inverse Laplace Transform of the function $$ F(s)=e^{c \cdot s^2} $$ where $c > 0$.
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How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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Transforming Exponential to Ordinary Generating Functions

I am looking for a particular ordinary generating function, if it exists for the Associated Stirling Numbers of the second kind $$b(1;n,j)=b(n,j)=\sum_{k=0}^j(-1)^k\binom{n}{k}S(n-k,j-k)$$ Where ...
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How to find Laplace transform of a differential equation?

$y′′ + 3y′ + 2y = f$ , $y(0) = 0$ , $y′(0) = 1$ where $f$ is given by $f(t) = \sum_{n=1}^\infty \delta(t−n)$; find a 1-periodic function $y_*$ with $\lim_{t\rightarrow \infty} |y(t)−y_*(t)| = 0$. I ...
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Heaviside function in the function whose Laplace transformation is $e^{-(\gamma+s)}/[(s+\gamma)^2+b^2]$

This is from a homework question 13.22 part (c) from "Mathematical Methods for Physic and Engineering" by Riley et. al on p. 464 I don't understand why the heaviside function is in the solution to ...
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The inverse Laplace transform of an entire function

A simple calculation shows that the Laplace transform of $f(t)=e^{-t^2/4}$ is the function $F(p)=\sqrt{\pi}e^{p^2}\operatorname{erfc}(p)$. I would like to find the inverse Laplace transform of ...
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Laplace transform on a finite interval $f(t)= \int_0^1 e^{-xt} f(x) \, dx$

What is the name of this transform? It's basically the Laplace transform where we integrate over a finite interval. $$ F(t)= \int_0^1 e^{-xt} f(x) \, dx$$ Is it still just the Laplace transform? ...
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Origin of Laplace Transform

Is the Laplace transform the continuous version of the infinite power series? $$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$ I learned this by watching this video lecture: ...
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Taking Laplace-Stieltjes transform to find virtual idle time in G/M/1 queue

I am reviewing some queueing problems from Gross and Harris, and had a question on problem 5.40 part b. The problem is stated as follows: Part B: Show that the stationary output of a $ G/M/1 $ queue ...
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The inverse laplace transform of $p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$ can be written in Fresnel integrals?

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
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Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
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Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
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Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
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Existence of the Laplace Transform

I'm seeing Laplace transforms for the first time, and I'm having trouble understanding the criteria for deciding when they exist. I've read a few websites and books that seem to say that we only ...
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How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
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Periodic Laplace transform

Here's the questions and the graph I've been struggling with this since Thursday and this is due today. I need help on problems a and b. For a, the question is: "Write $f(t) = \sum_{n=0}^\infty ...
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Criteria for $L^1$ convergence looking at Laplace transforms

Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ ...
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Find the inverse laplace transform: $\frac{1}{{{{({s^2} + 1)}^3}}}$

Find the inverse Laplace transform: $$x(t) = {L^{ - 1}}\left[ {\frac{1}{{{{({s^2} + 1)}^3}}}} \right]$$ with $x(t=0)=0$. I did: $${\left[ {{\mathop{\rm R}\nolimits} {\rm{es}}\frac{{{e^{st}}{{(s - ...
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Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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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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Borel-/Laplace-transform and $\psi$-function

I'm considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A (of course thought as of infinite size) $ \qquad $ The ...
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Find what values of 'b' have bounded solution(differential equation)?

$y′′ + b^2{y} = f(t)$ $ f(t) = t$ for $0 < t < 2\pi$ ($2\pi$ periodic sawtooth wave) This is my solution to the differential equation. $y(t) = C_1\cos(bt) + C_2\sin(bt) + b^{-3}\left(bt - ...
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Pollaczek-Khinchin formula for ruin probability - proof

I got stuck in a specific part of proof of the Pollaczek-Khinchin formula (in book "Stochastic Processes for Insurance and Finance", T. Rolski et al., section 5.3.3, theorem 5.3.4). Namely, why the ...
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Singularities of complex exponential and asymptotic expansion

Consider the equation $$L[u(x,t)] = \tilde u(s,t) = \frac{e^{-t\sqrt{s^2-1}}}{s-2}$$ I want to find $u(x,t)$ in the form of an integral. I first need to find the poles and singularities of the ...
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Inverse Laplace transformation correct?

I'm actually on the way to solve a little bit complicated differential-equation. Therefore I used the Laplace transformation. I've already solved it but I am actually not sure, whether my solution ...
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Solution of 1d wave equation by Laplace transform

This is a homework problem that I can almost finish. I just can't invert the Laplace transform at the end. $$u_{xx}=u_{tt}, u(t=0)=u_t(t=0)=0, u(x=0)=\sin\omega t, u(x=2)=0.$$ Taking the Laplace ...
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Computing transfer function then static gain

I am working on a problem for a flight controls class. I have an equation related to pure yaw. My goal is to get the transfer function associated with it, and then obtain the system static gain. The ...
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Why does the Laplace transform of $t^2 \exp(at)$ exist?

My book states a theorem : "Let $f(t)$ be a function piecewise continuous on $[0, A]$ for $ A > 0$ and have an exponential order at infinity with $|f(t)| \leq M \exp(at)$. Then the Laplace ...
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Laplace transform of 1

I am getting a little confused when using the Laplace Transform. So, taking the laplace transform of 1 we get: $\mathfrak L[1](p)= \int_0^\infty e^{-pt}dt=[-\frac1pe^{-pt}]_0^\infty=\frac1p$ when ...
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Laplace transform, Brownian semigroup

I have a question about the Laplace transform of a measure and a semigroup. Let $(\eta_{t})_{t>0}$ be a family of measures on $(]0,\infty[,\mathcal{B}(]0,\infty[))$. The Laplace transform ...
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All possible inverse laplace transform of $ (s+1)^2/(s^2+2s+1)$

Well this was the question given in our previous week's test: Find all possible Inverse Laplace Tranforms of $(s+1)^2/(s^2+2s+1)$ . I just expanded the numerator and so the function reduced to 1.And ...
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Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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Inverse Laplace Transform of a complicated function

Help with finding the inverse Laplace Transform of $$F(s) = \frac{1}{s\sqrt{M^{2}-s^{2}}}e^{\frac{\sqrt{N^{2}-s^{2}}}{s}}$$
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Laplace transform of $\sin(x)$

I am confused with Laplace transform of $\sin(\theta)$. For example, what is the LT of $A \sin(x(t))=Bx''(t)$ ($x$ is second order), $A,B$ are constants.
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Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
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Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
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Z - transform of a transfer function

I have to apply a z-transformation to my transfer function which looks like this: $$\frac{K}{s} - \frac{K\cdot T}{T\cdot s}+1$$ I have tried it and this is my result: $$K \cdot \frac{z}{z-1} - K ...
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Laplace transform, when $s \rightarrow \infty$

I'm reviewing lecture notes on Laplace Transform and there's one step that I don't understand: Find the solution to: $$x y'' + y' + xy = 0, y(0) = 1, y'(0) \mbox{ finite}$$ Taking the Laplace ...
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Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
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Help with an improper integral

Can someone please help me evaluate this improper integral? $$\int_{0}^{\infty}\exp\{-au^{-a}-u\}du$$ for $a>0.$
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Figuring out impulse response

I need a little help with figuring out this problem. I understand most of it but the main part I don't understand is: The signal $h''(t)+2*h'(t)+2*h(t)$ is of finite duration. In the problem we are ...
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Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with τ, λ, and k all real and ...
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The Laplace transform - does it have an associated differential operator, if the kernel is to be viewed as a Green's function?

I've begun learning about Green's functions, and if I understand correctly, the Green's function for a linear differential operator $L$ with appropriate boundary conditions is the kernel for the ...
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Transfer Function

Vehicle dynamic system (Bicycle model) is given by the following state space model (which also includes Road Bank angle): $\begin{Bmatrix}\dot{x_1}\\\dot{x_2}\end{Bmatrix}=\begin{bmatrix}a_{11}& ...