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

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closed form for the following integral which is similar to Laplace transform

I want to find a closed form for this integral: $\int\limits_{x=0}^{\infty} \exp(-\frac{1}{x})x^n\exp(-sx)dx$ I know that it has closed form for $n=0$ but what about $n\neq0$? Does anyone have any ...
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Inverse 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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Integrable slowly varying function

We say a function $L$ is slowly varying if $$\lim_{t\to\infty} \frac{L(tx)}{L(t)} = 1$$ for every $x > 0$. Are there such $L$ that are integrable? Say $L$ is defined on $[0,\infty)$ and is ...
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Inverse Laplace transform with branch cut

For the purpose of my research on persistent random walks I need to compute the inverse Laplace transform of $$ F(s)=\frac{\mathrm e^{-b\sqrt{s^2-1}}}{s^2-1}.$$ I looked up in tables of integral ...
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Nonstandard analysis and integral transforms

Can integral transforms be evaluated without limits(i.e Laplace transform) such as in non standard analysis? Can the improper integral be bounded by a hyperreal number? I am not very familliar with ...
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How to find the inverse Laplace transform of $\frac{s}{(s+1)^2(s+2)}$? [closed]

I would need a little help in finding the inverse Laplace transform of the function: $$f(s)=\frac{s}{(s+1)^2(s+2)}.$$ Thanks in advance.
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Why won't this Laplace transform work?

Solving an IVP: $$y'' - 2y' +5y = 0 $$ $$ y(0) = 2, y'(0) = 4$$ Taking the Laplace transform of both sides $$\mathcal{L}{y''} - 2\mathcal{L}{y'} +5\mathcal{L}{y} = 0$$ $$[YS^2 - S(y0) - y'(0)] - ...
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Which method solves this integral equation? $\int_{-1}^{1}w(x)\,e^{tx}\,dx=6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$

Today I encountered this integral equation wrt. $w(x)$: $$\int_{-1}^1 w(x)\ e^{t x}dx = 6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$$ I never solved such equations, and when I tried to ...
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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. ...
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Help with a particular Ordinary Differential Equation

I did the following problem but I am coming up with the wrong answer. Problem: Use Laplace transforms to solve the following system. All unknowns are fuctions of $x$. \begin{eqnarray*} w'' + y + z ...
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building a laplace transform

F(s) is the laplace transform of the function (f(t)) Numerator is equal to 1 and the denominator is a 2nd degree polynomial with complex conjugate roots of my choice (both s and w are non zero in s ...
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Transfer Function, relation of numerator/denominator to output/input

simple transfer function: $ \frac{Y(z)}{U(z)} = \frac{B(z)}{A(z)} $ from my point of view the Y(z) is related to B(z) and U(z) is relater to A(z). The lecture notes says, it is the other way. the ...
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inverse laplace transform of a transfer function

So I'm working on this problem but the $$e^{-s}$$ term is throwing me off.. $$ G(s) = \frac{100(s+2)}{s(s^{2}+4)(s+1)}e^{-s} $$ Can someone help me out? I tried using partial fraction expansion to ...
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70 views

Cauchy's theorem and a contour integral

$$\frac1{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} ds \, \frac{e^{-\sqrt{a s}}}{c s+s^{3/2}} \cos{\sqrt{b s}} \, e^{s t}$$ where $a, b$ and $c$ constant. To evaluate this, I used Cauchy's ...
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33 views

Finding inverse Laplace transform of a fraction of polynomials

I am trying to find the inverse Laplace transform of $$\frac{4s^3 + s}{s^2+1}$$ I tried polynomial long division and reduced it to the following expression: $$4s - \frac{3s}{s^2+1}$$ But I'm not ...
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Compute laplace transform of $\frac{\cos\sqrt t}{\sqrt t}$?

What is the Laplace transform of $\frac{\cos\sqrt t}{\sqrt t}$?
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Inverse Laplace Transform of $F(s) = \frac{3s+8}{(s^2+2s+20)^2}$

Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor $$F(s) = \frac{3s+8}{(s^2+2s+20)^2}$$ tried to apply partial fractions to it and i just ...
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54 views

Calculate the inverse Laplace transform of $\frac{1}{1-e^{-s}}$

During my signals and systems class i came across this and i have to find it's inverse Laplace transform. I don't know how. $$\mathcal{L}^{-1} \Big\{ \frac{1}{1-e^{-s}} \Big\} = \ ?$$ Any help ...
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Understanding the Bromwich Integral (Inverse Laplace Transform)

The formula for the Inverse Laplace Transform is (Bromwich Intergal): $$f_{(t)}=\frac{1}{2\pi i}\lim_{x\to\infty}\int_{\alpha-x i}^{\alpha+x i} \left(e^{st}\cdot F_{(s)}\right) \text{d}s$$ My ...
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Conditions of an inverse Laplace Transform

$$\mathcal{L}(c)=\int_{0}^{\infty} c\cdot e^{-st}\text{d}t=\frac{c}{s},s>0$$ $$\mathcal{L}^{-1}\left(\frac{c}{s}\right)=c$$ What are the conditions to the last inverse Laplace Transform?
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Conditions to have an inverse laplace transform?

If I had a function $\widehat f(s)$ how would I know if there exists a function $f(t)$ so that the laplace transform of $f$ is $\widehat f$? From looking at the formula for finding the laplace ...
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Transfer function from S-space to Z-space

simple question: I got: $$ G(s) = (\frac{s^2 + 1}{s - 1}) $$ I am supposed to get directly to Z-space with sampling period as parameter h. So far I have tried to divide the function into: $$ G(s) ...
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Simplification of rational exponential integral

I can't figure out how this sort of simplification works, could someone explain the process behind this? $$\oint \frac{1}{2\pi i} \frac{e^{st}}{(s+a)(s+b)}ds$$ $$=\oint \frac{1}{2\pi ...
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Solving system of 1st order ODE's using Laplace transforms - stuck on algebra

Here is the system: $$x'(t) + y'(t) + x(t) + y(t) = 1$$ $$y'(t) - x(t) + y(t) = -t$$ I simplified this to the following system of simultaneous equations, with ${\scr L}[x] = X$ and ${\scr L}[y] = Y$ ...
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How to calcute the inverse Laplace transform of $\hat{F}(z)=\sum_{i=0}^{\infty} \frac{A^{i}}{z^{i+1}}$

I am reading a paper which in part of that authors used to calculate the inverse Laplace transform in a way that I can not understand. actually suppose we have $ \hat{F}(z)=\sum_{i=0}^{\infty} ...
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Differentiation of Laplace Transform

It is known that The $s-$derivative rule states that $$ \mathcal{L} (t^{n} f) = (-1)^{n} F^{(n)} (s) $$ The proof for the laplace differentiation involves \begin{align*} F'(s) &= ...
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limitation of initial value theorem

I am student and stuck in this question , this question was asked to me on exam , what is the limitation of initial value theorem ,but i was not able to think of limitation Since the time was running ...
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ODE using Laplace transform

[ I got my Y(t) to be : $$12 \, e^{-4} \, e^{-2s} \, [\frac{1}{12(s+2)} + \frac{1}{4(s-2)} - \frac{1}{3(s-1)}] + \frac{1}{(s-2)} - \frac{1}{(s-1)}.$$ so i assume I need to use t shifting for the ...
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Choosing between SOV/Green's functions/Laplace transform for solving PDE - Guideline for choosing the most appropriate method?

Forgive me if this questions seems silly, but I have a question which is keeping me busy. I'm not really looking for a mathematical proof (but it is welcome), however I'm more looking for guided ...
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Still getting wrong answer after trying to solve $x''(t)+4x(t)=t^2$ where $x(0)=1$ and $x'(0)=2$

I am trying to solve this differential equation: $$x''(t)+4x(t)=t^2,x(0)=1,x'(0)=2$$ The answer should be: $$x(t)=\frac{1}{4}t^2-\frac{1}{8}+\frac{9}{8}\cos{2t}+\sin{2t}$$ Which is also verified ...
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Inverse Laplace transform of a product of integrals

I'm struggling in demonstrating that the relation in the first quoted equation. I've met this problem in finding the inverse Laplace Transform of an equation in the form $W=A\times B$, there $A$ and ...
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Solving general linear ODE $\sum_{k=0}^n y^{(k)}=0$

Is there a way to solve this general linear ODE: $$\sum_{k=0}^n y^{(k)}=0$$ For the first few $n$ here are the solutions: $$\begin{array}{c|c} n & y \\ \hline 0 & 0 \\ 1 & c_1 e^x \\ 2 ...
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Help with two functions - continuity, Laplace transform and Fourier series [closed]

I've been practicing for my exam lately, and there are two function that I've had a real trouble analyzing. 1.$f(x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{10^n \sin(x)}$, for $x \neq k\pi$ $f(x) = ...
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Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
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70 views

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 ...
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Laplace vs Fourier density representation of a positive rv

Given a general random variable $X$ with density function $f(x)$ and characteristic function $\phi_X(u)$ we can go back and forth between the density and the characteristic by using the Fourier ...
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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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To find Inverse Laplace of $\,F(s)=\log\dfrac{s+1}{(s+2)(s+3)}$

To find Inverse Laplace of $$F(s)= \log\frac{s+1}{(s+2)(s+3)}.$$ I have tried to use shifting theorems, but of no use. Should I apply series for log and take inverse laplace of individual terms, if ...
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How to find laplace transform of $\,\sinh(ct)\int_a^te^{au}\sinh(bu)\,du$

How to find laplace transform of $$\sinh(ct)\int_a^te^{au}\sinh(bu)\,du.$$ I tried to integrate inner function and then do it, but it became way more tedious. So I was thinking there should be good ...
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Inverse Laplacetransform of rational function with multiple pole

I have to calculate the inverse Laplacetransorm of this function using Residue calculus $$ \frac{s^4 + 6s^3 - 10s^2 + 1}{s^5} $$ but I can't find any Residue rule that would solve this. Can you show ...
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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 $-\infty<x<\infty$ , ...
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matrix elementary column operations

Till now i was using the elementry row operations to do the gaussian elemination or to calculate the inverse of a matrix. As i started learning the Laplace's transformation to calculate the ...
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Two different answers with Laplace

Find the solution for the equation $$ -u'' + u = \delta'(t)$$ for which it "disappears" for $t<0$ By using residuals! So I used Laplace transformation for this. $$Y(-s^2 + 1) = ...
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Solving Second Order PDE with Dirac Delta

I want to find the functional form of the Green function G(x,t) for a parabolic differential equation: $$ ...
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Convolution of two piecewise functions using Laplace transform [closed]

I'm practicing Laplace transforms and I stumbled upon one question which I am not exactly sure how to tackle. The question is: Using Laplace transforms (or otherwise) calculate the convolution of ...
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40 views

Solve 2nd order ODE using Laplace transform

Im trying to solve a laplace transoform question, but i am stuck. The question is $y''(t)+2\zeta y'(t)+y(t)=0$,$y(0)=1$,$y'(0)=0$ and $\zeta=2$. I have so far done: Laplace transform which gives ...
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63 views

Where is my error in solving $y'' + y' + y = 0, y(0) = 1, y'(0) = 0$ with Laplace transform?

Im trying to solve a laplace transoform question, but i am stuck. The question is $y′′(t) + 2ζy′(t) + y(t) = 0, y(0) = 1, y′(0) = 0$ and $ζ = 0.5$. I have so far done: Laplace transform which gives ...
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Scaling property of Laplace transform

I am not sure how to do the following problem: Let $$\hat{F}(s)=\mathfrak{L}(f(t))$$ be the Laplace transform of $f(t)$. Show that: $$\mathfrak{L}(f(at))=\frac{1}{a}\hat{F}\left(\frac{s}{a}\right) ...
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A very simple question: what spaces of function does the Laplace transform map from and into?

Given a function $f$, we can write $f\colon\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the Laplace transform operator ...