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

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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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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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62 views

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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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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36 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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31 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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55 views

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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1answer
28 views

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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43 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. $$L\{y''+3y'\}=L\{f(t)\}$$ $$s^2Y-sy(0)-sy'(0)+3(sY-sy(0))=L\{t\}+L\{1\}-L\{u4(t)(t-4)\}-5L{u8(t)}$$ ...
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How to compare ZOH and tustin

I'm discretizing some continuous time systems. Now there (MATLAB) are of course different types of discrtization methods, among them tustin (bilinear), euler backwards, euler forward etc. Often one ...
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Unilateral Laplace transform calculation

I'd like to verify that $\mathcal{L}[e^{-at}]=\frac{1}{s+a}$, $t\ge 0$. So I calculate: $$\int_{0^+}^{+\infty} e^{-at} e^{-st} dt=\int_{0^+}^{+\infty} e^{-t(a+s)} dt = \frac{1}{-(a+s)} ...
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Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( 1+\frac{w^{2}}{s^{2}}\right ) \right \}$

Where $s\in \mathbb{C}$. I assume that this would be pretty easily handled by solving it by definition, but I haven't taken courses in complex analysis yet. Also, I can't think of any nice property of ...
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transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
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The Laplace transform of $\exp(t^2)$

A naive attempt to calculate the Laplace transform of the function $f(t)=e^{t^2}$ results in integrals of the form $$\int_0^\infty e^{t^2-st}dt,$$ which obviously don't exist as the integrand grows ...
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Unilateral Laplace transform

I tried to do the same unilateral Laplace transform in two ways, but I got different results. I have to calculate: $\mathcal{L}[r(t-1)]$, where $r(t)$ is the ramp function, that is $r(t)=t, t\ge0$. ...
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Find the Laplace Transform

Could anyone enlighten me on how to find the Laplace Transform of $$\frac{1-\cos (t)}{t}$$
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Inverse Laplace Transform of $1/(s+1)$ without table

The pole is on the left half plane, so $\gamma =0$ $$\frac{1}{2i\pi}\int ^{i\infty}_{-i\infty}\frac {e^{st}}{s+1}ds$$ substituting $iu=s$ $$\frac{1}{2i\pi}\int ^{\infty}_{-\infty}\frac ...
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How do you find the inverse Laplace transform of $\frac{1}{\sqrt{s}(s-a)} $

When I use the convolution method, I can't avoid getting a divergent integral.
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Laplace of $\int_0^t \frac{sinx}{x}dx$

What is the Laplace transform of $\int_0^t \frac{\sin x}{x}dx$ I'm thinking about approaching it as a convolution but I am not sure how. Could I define it as the convolution of $1$ and $\frac{\sin ...
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System of differential equations using Laplace transform

Using Laplace transform, solve the system: $w'+y=\sin(x)$ $y'-z=e^x$ $z'+w+y=1$ where $w(0)=0$ and $z(0)=y(0)=1$.
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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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Relationship between Inverse Fourier and Inverse Laplace Transform?

Suppose we are given a fourier transform $$ F(\omega) = \frac{1}{\omega^2+4} $$ Can we use inverse laplace tranform by taking $i\omega = p$ to find the inverse fourier transform? I did this and got ...
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Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, ...
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Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
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A proof which results in Gamma (or Erlang) distribution-From Karlin & Taylor's “A First Course in Stochastic Processes”

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
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laplace step function $H(π-t)(\sin(t))^2$

How to calculate the laplace transformation of $H(π-t)(\sin(t))^2$ ? I know that I have to use $\sin^2(t)= 1/2(1-2\cos(2t))$ but i am stuck of how to proceed``
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Find the Laplace Transformation of $H(\pi-t)$.

I know how to find the Laplace Transformation of $H(t-\pi)$, but what about if the $t$ is negative. Any help is much appreciated.
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When does Fourier Transform be the same as Laplace's?

I have the TI nspire CX CAS... it can perform Laplace Transform but can't perform Fourier Transform. They are equal in some problems, but not all the time! So, when does both of them be equal so that ...
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Laplace transform (Simple factorization)

The question require me to find the inverse of Laplace transform. In the first line of solution, how does it go from LHS to RHS? Does it simply apply partial fractions?
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Solving initial value problem using Laplace transforms, one other method, and comparing results

So for my solution using characteristic equations I get (fixed a typo for first coefficient) $$\frac{11}{30} e^{-3t} - \frac{21}{20} e^{-2t} + \frac{21}{20} e^{2t} - \frac{11}{30} e^{3t}$$ For the ...
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Evaluating the inverse Laplace transform of $1/(s^2-\sum_{n=1}^\infty{n!s^{3-n}x^n})$

I want to evaluate at $t=1$ the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of $$ F(s) = \frac{1}{s^2-\sum\limits_{n=1}^\infty{n!s^{3-n}x^n}} $$ and find out the $x^n$ ...
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Are there functions that are not of exponential order for which you can define a Laplace transform?

I'am in a course of Introduction to Linear Differential Equations and teacher made us this question in class. we work in $\mathbb{R}$, and any help to answer this is welcome
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determine the time domain equation of the output responseusing inverse laplace transform, given a step input

I have this initial transfer function \begin{equation*} \frac{Y_s}{F_s}=\frac{1}{(1s^2+2s+2)} \end{equation*} unit step is $F_s=\frac{1}{s}$ so I then get \begin{equation*} \frac{1}{s(s^2+2s+2)} ...
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Find the roots of the corresponding characteristic equation

The equation is $${Y_s\over F_s}={1\over s^2+2s+2}$$ I have got to $$r^2+2r+2=0$$ what do i need to do next?
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Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
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solving second order linear differential equation

Can somebody please show me how to solve the following differential equation: $$ a\ddot{x} + b\dot{x} = c $$ given these initial conditions $x(0) = 2$, $\dot{x}(0) = 0.5$ and $a = 4, b = 1.5$ First ...
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Laplace transform and “imaginary infinity”

I was recently studying Laplace transform for the first time, and I'd like to ask the following thing: there was an integral with limit of integration, something like that: a+j×infinity, j the ...