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

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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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Integral of Absolute Value of $\sin(x)$

For the Integral: $\int |\sin (ax)|$, it is fairly simple to take the Laplace transform of the absolute value of sine, treating it as a periodic function. $$\mathcal L(|\sin (ax)|) = ...
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The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
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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 derivative squared

I'm trying to solve a problem while I'm studying Control Theory and I came up with a difficult question. $ \mathcal{L}\left[y'(t)^2 \right] $ Basically I need to find the Laplace Transform of this ...
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“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
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What is the inverse laplace transform of $\large{\frac{ s^3 - a^2s }{(s^2 + a^2)^2}}$

I tried convolution and partial fractions but both turned out to be too much work. Is there any easy work around??
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Laplace transfer function and quasi-sinusoidal input

Let's suppose we have an LTI system whose Laplace domain transfer function is: $$ F(s)=\frac{1}{s^2 + \frac{\omega_y}{Q_y}s + \omega_y^2} $$ Its input is the Coriolis force. Such force is experienced ...
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Instantaneous DC output using Laplace transform

I am testing a feedback system in which a feedback signal from sensor is applied to a correction network and then compared with threshold values. I need some clue how to achieve this, I will be ...
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Convert an equation in Laplace “s” space to Discrete “z” space using a table

I'm trying to discreteize an equation. I have the equation in laplace form, but I do not have the original differential equation. The equation is: $$\frac{\theta(s)}{V(s)} = \frac{a}{s(s+b)(s+c)}$$ ...
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Evaluate $\int_{0}^{\infty}\frac{\sin^{2}\left ( t \right )}{t^{2}}dt$ with help of Laplace transform

Using the following identity $$\int_{0}^{\infty}\frac{f\left ( t \right )}{t}dt= \int_{0}^{\infty}\mathcal{L}\left \{ f\left ( t \right ) \right \}\left ( u \right )du$$ I rewrote ...
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Transform function in terms of Heaviside functions.

I am working on the red-shaded problem below however I am unsure how to use my answer from part b to answer the question. Any help is appreciated.
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1answer
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Initial Value Problem involving Laplace Transforms

I am trying to solve the initial value problem: $y' + 5y = $ 0 if $t \in [0,3]$ 9 if $3 \in [3,6)$ 0 if $t \in [6, \infty)$. with $y(0)=9$. I am asked to take the Laplace Transform of both sides ...
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Laplace Transform of $e^{2t-12}u(t-6)$.

I am trying to find the Laplace Transform of $e^{2t-12}u(t-6)$. All I know is that $\mathcal{L}\{e^{-at}\} = \dfrac{1}{s+a}$ and that $\mathcal{L}\{u(t-a)\} = \dfrac{e^{-as}}{s}$. I also know that ...
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Using the Heaviside function to represent a given graph

The question is the following: The graph is zero between 0 and 2, is a straight line from the point (2,0) to (5,5), a straight line down from (5,5) to (4,0) and zero everywhere else. So far, my ...
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How to explain this simplification here?

I can't understand this simplification the book says without explanation. Could someone help me? It is the calculatation/development of the transfer function of a digital system composed by a dac ...
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An initial-value problem and a corresponding Laplace Transform

I am to solve the 1st-order ODE \begin{align*}y^\prime + 3y = 45t, \qquad y(0) = 6 \end{align*} using a Laplace Transform for a problem set. So far, by taking the Laplace Transform of both sides, I ...
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Inverse Laplace Transform of $\dfrac{6s -19}{s^2 - 6s + 13}$.

I am trying to figure out the inverse laplace transform of $\dfrac{6s -19}{s^2 - 6s + 13}$. Looking at my table of Laplace Transforms in my textbook, it seems that either I must break up this fraction ...
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What's the logic in this equation? [laplace and z transform] [closed]

Why/How the z^{-l} goes there in 3.25? Also I can't understand the 3.26 statement.
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48 views

Initial Value Problem using Laplace Transforms

Solve using Laplace Transform: $$y''(t)+2y'(t)+5y(t)=xf(t), \\ y(0)=1,y'(0)=1$$ where $x$ is a constant. Once the solution is found, evaluate the limit as $t \to\infty$. Progress: If I have ...
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39 views

How to solve an intro-differential equation with integrals.

$y'(t)-y(t)-3\cdot\int[e^{x-t}\cdot y(x),x,0,t]=16\cdot t, y(0)=16$ I am having a difficult time figuring out how to evaluate the integral to solve with the rest of the problem.
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How to solve an integral with an exponential and function.

How do I solve the integral $ \int \limits _0 ^t e^x y(x) \Bbb d x$? I know the integral of $y(x)$ would result in $\frac {Y(s)} s$.
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What are the deduction of $\sin\left(\omega_0t+\theta\right)$ for Laplace form?

I have the following function in time domain $\sin\left(\omega_0t+\theta\right)$, which is $\left(\frac{s\sin\left(\psi_1\right)+\omega_0\sin\left(\psi_1\right)}{s^2+\omega_0^2}\right)$ in Laplace ...
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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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What is the Laplace Transform? [duplicate]

What is the Laplace transform? I understand how to do it (taken differential equations), but my professor just kinda told us to accept that $ F(s) = \int_0^\infty e^{-st}f(t)dt $ is gospel and to ...
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1answer
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Reason for $0^-$ bound on unilateral Laplace transform

I have the definition of the unilateral $\mathcal{L}$-transform, valid for causal signals, to be: $$\mathcal{L}\left[f(t)\right](s)=\int_{0^-}^{+\infty}f(t)e^{-st}dt$$ My question is regarding the ...
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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 ...
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Why is Laplace transform so useful

I recently encountered the term Laplace transform. Since My background is mathematical, the definition itself is very easy for me. The thing which is less clear for me is the context. I assume, that ...
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Numerical or analytical or exisistence: Inverse Laplace Transform

Edit 1: With the hint of Ron, we can simplify the question to : $$\bar{f}(s)=\frac{1}{(s^2+1)\arctan s }$$ So what about this function's inverse Laplace Transform? Or can anyone tell me that the ...
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Laplace transform,Fourier transform and Z transform mathematical equations

Fourier transform $x(w)$ of signal x(t) is given by $$ x(w) = \int\limits_{t=-\infty}^{+\infty} x(t) e^{-j w t} dt -(1)$$ Laplace transform $x(s)$ of signal x(t) is given by $$ x(s) = ...
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Proof of a Bromwich integral formula

I am trying to prove that: $$\frac{1}{2\pi i }\int_{\alpha-i\infty}^{\alpha+i\infty}\frac{(\log s)^{n}}{s}e^{sx}ds=(-1)^{n}\frac{d^{n}}{dz^{n}}\frac{x^{z}}{\Gamma(1+z)}\left.\begin{matrix} \\ \\ ...
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55 views

Conditions for existence of inverse Laplace transform.

Given a function $F(s)$, how to check if inverse Laplace transform of $F(s)$ exists? In other words, I want to know conditions for existence $f(t)$ such that $$ \int_0^\infty e^{-st}f(t)\,ds = F(s) ...
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Proving an equation involving integrals and limits

I have to show the following equation: $\large\int_0^\infty \! e^{-st}\cos(\beta t) \, \mathrm{d}t=\frac{s}{s^2+\beta ^2}$ with $s>0$ I've come so far: $\large\int_0^\infty \! e^{-st}\cos(\beta ...
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Interchanging Inverse Laplace Transform

I have a function $f(|\boldsymbol{k}|,s,\theta)$ for which I am interested in its inverse Laplace transform. I am also interested in the function's mean value for constant $|\boldsymbol{k}|$, but ...
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60 views

Laplace transform of function

Assume that $f(u)=(\frac{b}{πu^3})^{1/2} e^{2b} e^{-bu} e^{-b/u}$, where $b>0.$ I am trying to calculate the Laplace transform $L\{f(u)\}(s)$ and then the $n_{th}$ derivative of this transform, ...
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26 views

Inverse Laplace of a function

I am really searching for hours now for the inverse laplace transformation of the following function: $$\frac{75s + 12739.726}{s( 0.0365s^2 + 81.2s + 12739.726)}$$ If I put this in WolframAlpha the ...
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35 views

Laplace transform of inverse gaussian distribution [closed]

Can someone write in details how i can derive the Laplace transform of the Inverse Gaussian distribution? I think i am missing something during the calculation of the interval which gives the Laplace ...
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1answer
24 views

What does it mean “Laplace transformable functions”

I am reading about the The convolution operation, and the notion Laplace transformable functions is mentioned there. Doe anyone know what is the definition of Laplace transformable functions? Thank ...
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24 views

Inverse Laplace transform with minus $\Delta$ in denominator

Please help me find this inverse Laplace transform. $$ F(s)=\dfrac{2s-3}{s^{2}-2s+2} $$ I couldn't resolve the denominator, because the quadratic has discriminant $\Delta=-4$.
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Use of Laplace transform to solve initial value problem.

--Short Explanation: I have to say I am going crazy with this problem as it does not give me the same as the suggested solution in the book: Problem: $y''-7y'+10y=9\cos{t}+7\sin{t}$ $y(0)=5$, ...
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Laplace transform of the wave equation

I started of with the wave equation $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$ with boundary conditions $u=0$ at $x=0$ and $x=1$ and initial condition $u=sin(\pi x)$ and ...
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Laplace transforms to solve heat equation

I have the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ Boundary conditions are $u=0$ at $x=0$ and $x=1$ The initial condition is $u=\sin(\pi x)$ I know that $$L ...
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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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Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
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For Laplace Transforms; What is the interpretation of $s$ compared to $t$? Why is each Laplace transform only defined for some values of $s$?

What is the interpretation of $s$ compared to $t$? Why is each Laplace transform only defined for some values of $s$?
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use laplace transform to solve the given integral equation

use Laplace transform to solve the given integral equation I don't know how start because it differences on other Laplace question I see before
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Can this definite integral of an inverse Laplace transform by simplified?

Can either of the below expressions involving an unknown analytic function $h(s,t)$ and the inverse Laplace transform $\mathcal{L}^{-1}$ be simplified? $$ \int\limits_{0}^1 \mathcal{L}^{-1} \left\{ ...
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1answer
41 views

Laplace diffrential equation

$$\frac{dx}{dt}=2x +3y$$ $$\frac{dy}{dt}=3x +2y$$ Find general solution. I know there is a solution through eigenvalues. But I want to solve it with Laplace transformation. I almost get the right ...
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The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
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Laplace Transforms

Solve the initial value problem for y(t) using Laplace Transforms. $$L\{y''+3y'\}=L\{f(t)\}$$ $$s^2Y-sy(0)-sy'(0)+3(sY-sy(0))=L\{t\}+L\{1\}-L\{u4(t)(t-4)\}-5L{u8(t)}$$ ...