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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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} ...
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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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Doubts relating to Spaces of type $\mathcal{S}$

I have doubts in the following two questions : 1) What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , ...
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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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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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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 ...
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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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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) = ...
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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 ...
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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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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 ...
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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 ...
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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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Probability proof of inversion formula for Lapalce transform

Let $f:[0, \infty[\longrightarrow \mathbb{R}$ be bounded and continuous and define $L(\lambda)=\int_0^\infty e^{-\lambda x}f(x)dx$. Let $X_n$ be a sequence of independent random variables with ...
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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 ...
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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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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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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 ...
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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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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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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 ...
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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 a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
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Inverse Laplace Transform with a functional

I don't know if it is possible, but I would appreciate if someone help me to obtain the inverse Laplace transformation of the function $F(s)=hs/(s^2+w(s)^2)$. Where $s$ is a complex number and $w$ is ...
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Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
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A discussion on fourier and laplace transforms and differential equations …?

i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of ...
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What is the mapping of Z-transform?

Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as: $$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real ...
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What does it mean to take the Laplace transform of a non-periodic position function?

What I'm trying to get through my head here is how taking the Laplace transform of a system with a position function like $X(t)=t$ is possible. To my current (admittedly incomplete) ...
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Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
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Reference for an identity from Abramowiz and Stegun

I am curious as to where this identity was originally obtained. Any suggestions? $$ \frac{1}{\mathop{\Gamma}\nolimits\!\left(1+2\mu\right)2\pi i}\int_{-\infty}^% ...
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solving improper integral through laplace transform

My Problem is : Evaluate $\int_0^{\infty} \sin(t^2)dt $ using laplace transform: can some one give me hint to solve this.
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Inverse of Mellin transform with lower bound at $1$

I've seen two definitions of the Mellin transform: more commonly, $g$ is the Mellin transform of $f$ if $$g(s)=\int_0^\infty x^{s-1}f(x)\; dx,$$ or secondly, and more rarely, defined by the same ...