The Laplace transform is a widely used integral transform (transformation of functions by integrals), 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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The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ $$\...
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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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Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} e^{st}\frac{\Omega^2}{(s^2+4\...
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zeros/poles of Laplace transforms of Dirac combs (Riemann zeta function)

let's define $p_\alpha(n) = \displaystyle\int_1^n x^\alpha dx$ so that $\left\{\begin{array}{lll} p_0(n) &=& n-1 \\ p_{-1}(n) &=& \ln n \\ p_\alpha(n) &=& \frac{\textstyle n^...
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Laplace transform of functions related to type $\mathcal{S}$, and the relation to entire functions

I have doubts in the following two questions : What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , $k,q=0,1,2,......
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Is the Laplace transform a vector space isomorphism? And what space is it isomorphic to?

The laplace transform is a linear transformation, $\mathcal{L}: \mathcal{M} \rightarrow?$, where $\mathcal{M}$ is the set of exponentially bounded functions on $\mathbb{R},$since $\mathcal{L}(af(x)+bg(...
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inverse Laplace transform by finding residues of essential singularities

I want to find the inverse Laplace transform of $$F(s)=\exp\Big(-\sqrt{2s}\tanh(\sqrt{2s})\Big).$$ Despite the square roots, $F$ doesn't have any branch points since $$\sqrt{2s}\tanh(\sqrt{2s})=\frac{\...
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Inversion of Laplace transforms - simplifying the Bromwich integral

I have trouble following the derivation of equation $(2)$ in this paper. The authors define the Laplace transform of a real-valued function $f(t)$ of a positive real variable $t$ as \begin{equation} ...
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riemann zeta function : entire and even Laplace transforms

$$\xi(s) = s(s-1)\pi^{-s/2}\Gamma(s/2) \zeta(s)$$ $$\xi(s) = \xi(1-s)$$ thus $\Xi(s) = \xi(1/2+s) = \Xi(-s)$ is even, and furthermore it is an "entire and even Laplace transform" : $$\Xi(s) = \int_{...
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Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
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Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
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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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Why Fourier transform is not sufficient and we have to use Laplace transform?

Is there an easy way to explain the motivation behind the use of Laplace transform instead of Fourier transform? Isn't that any periodic function can be represented by sines and cosines? Why to ...
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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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On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right) $

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
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Tauberian theorems in queing theory

I'm trying to use Tauber's theorem below (Feller 1971, chapter XIII.5) "Let U be a measure with a Laplace transform $\omega(\lambda)$ defined $\forall \lambda >0$ and $t,\tau>0$ s.t. $t\tau=1$, ...
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How and why an integral Transform is created?

I don't know if what I'm going to ask will make any sense, but I was just wondering about integral transforms. I am talking about, for example, Mellin Transform, or Laplace Transform or Hilbert ...
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Application of Laplace transform to stopping times and expectations

Let $X_k$ be i.i.d. random variables such that $E[X_1]=m<\infty$. Consider $S_n = \sum_{k=1}^{n} X_k$. Let $\tau$ be a stopping time independent of $X_k$ with respect to the filtration $\{F_n\}_{n \...
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Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse transformation?...
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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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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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Methods for solving nth order semilinear elliptic PDEs

I am looking for names of methods, and examples of their use that can be used to find solutions for semilinear elliptic PDE equations of the below types: $$\frac{\partial^ny}{\partial x^n}+\frac{\...
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Analysing the modes of a signal with Laplace transform

If I have a linear dynamical system (assume continuous time for the time being) I can create the transfer function, let's say: $$\frac{1}{(s+a_1)(s+a_2)}$$ and the pole-zero map (this one is for e.g. ...
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Inverse Fourier transform using laplace

We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they ...
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Stopping time distribution and transforms with 1-dimension B-motion.

Let $W_t$ be a 1-dimensional Brownian Motion. For $x>0$, we define: $$\tau_{x} = inf \{ t \geq 0; |W_t| = x\}$$ Compute $E[e^{-s\tau_x}]$ and prove that $\tau_x$ is equal in distribution to $x^2\...
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Inverse Laplace transform of a hypergeometric function

I managed to solve an initial value problem in the Laplace domain in terms of a special function $ F(s) = c_2 \frac{1}{{{\left( {{s}^{1 +\beta}}-1\right) }^{\frac{1}{\beta+1}}}}+ c_1 \frac{s}{{{\left(...
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How to evaluate Integro-Differential Equation using Laplace convolution?

Can someone please explain how I begin to evaluate the following integro-differential equation? I know that it involves a convolution, but the $y(τ)$ within the integral is throwing me off. $$\int_0^...
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Why does the imaginary part of $s$ have no effect in analyzing region of convergence for Laplace Transform?

The tutorial that brought this assertion to me was: http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node2.html "As the imaginary part $\omega=Im[s]$ of the complex variable $s=\sigma+j\...
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Laplace transform of inverse error function

I want to calculate the convolution of a function with the inverse error function. Therefore I chose to try to first find an integral transform of the inverse error function like the laplace transform:...
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Asymptotics of Laplace transform at minus infinity

I am interested in relating the asymptotic behavior of a function $f(t)$ for large values of $t$ with the asymptotic behavior of its Laplace transform $\hat{f}(s)$ for small values of $s$. In practice ...
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For what types of differential equations is the Laplace transform most effective?

I'm reviewing for a final exam and want to make sure I know what tools to use for what situations, and was just wondering if there were situations where the Laplace transform is unusable or less ...
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Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
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Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) &...
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Translated Laplace transform

Is there any way to rewrite the Laplace transform is such a way that that one can apply to an IVP not centred at zero, that is, at some $y^{(n)}(a_n) = b_n$ for $n\in\mathbb{N}$ and $a_n \in\mathbb{R}\...
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Regarding the unilateral Laplace transform of LTI systems

Consider an LTI system described by the following differential equation, $$ \sum_{k=0}^{N}a_k\frac{d}{dt^k}y(t) = \sum_{k=0}^{M}b_k\frac{d}{dt^k}x(t) $$ With initial conditions, $$ y(t)|_{t=0}, \...
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Is it feasible to think of laplace transform and z transform as projections?

For Fourier transform, it has been ingrained in my head that all we are doing is projecting a function onto its Fourier basis, namely $(1, cos(t), sin(t),...cos(nt), sin(nt) ...)$ Can anyone comment ...
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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 $S(n,...
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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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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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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 $F(p)$....
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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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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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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 $V(r,z)$...
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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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Laplace transformation of $\frac{\cos x}{x}$

How would you find the laplace transformation of $\frac{\cos x}{x}$? Like I know you need to add it to another transformation to solve for it. And I know that the transformation of $\cos x$ is $\frac{...