The Laplace transform is a widely used integral transform, 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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Inverse Laplace Transform of …

I am trying to find the inverse Laplace transform of $$\frac{1}{pe^{p}}\int_2^p\frac{e^{q}q}{q-1}dq.$$ This function decays as $p$ goes to infinity, so it's reasonable trying to find it's Laplace ...
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Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
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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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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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Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
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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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inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

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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Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
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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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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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283 views

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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557 views

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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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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A rational integral with exponential denominator

Prove that: $$\int_{-\infty }^{+\infty }{\frac{{{x}^{4}}\text{d}x}{\left( \beta +{{\text{e}}^{x}} \right)\left( 1-{{\text{e}}^{-x}} \right)}}=\frac{\left( {{\pi }^{2}}+{{\ln }^{2}}\beta ...
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Laplace transform of the square root of a generic function

Let $f(t)$ be a function (for example of time $t$). Is there a general expression of the laplace transform of $\sqrt{f(t)}$ ? Same question for the inverse Laplace transform : Let $f(s)$ be the ...
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Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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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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Laplace transform of $\sin(x)$

I am confused with Laplace transform of $\sin(\theta)$. For example, what is the LT of $A \sin(x(t))=Bx''(t)$ ($x$ is second order), $A,B$ are constants.
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Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
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Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
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Z - transform of a transfer function

I have to apply a z-transformation to my transfer function which looks like this: $$\frac{K}{s} - \frac{K\cdot T}{T\cdot s}+1$$ I have tried it and this is my result: $$K \cdot \frac{z}{z-1} - K ...
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Laplace transform, when $s \rightarrow \infty$

I'm reviewing lecture notes on Laplace Transform and there's one step that I don't understand: Find the solution to: $$x y'' + y' + xy = 0, y(0) = 1, y'(0) \mbox{ finite}$$ Taking the Laplace ...
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Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
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Help with an improper integral

Can someone please help me evaluate this improper integral? $$\int_{0}^{\infty}\exp\{-au^{-a}-u\}du$$ for $a>0.$
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Figuring out impulse response

I need a little help with figuring out this problem. I understand most of it but the main part I don't understand is: The signal $h''(t)+2*h'(t)+2*h(t)$ is of finite duration. In the problem we are ...
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The Laplace transform - does it have an associated differential operator, if the kernel is to be viewed as a Green's function?

I've begun learning about Green's functions, and if I understand correctly, the Green's function for a linear differential operator $L$ with appropriate boundary conditions is the kernel for the ...
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Transfer Function

Vehicle dynamic system (Bicycle model) is given by the following state space model (which also includes Road Bank angle): $\begin{Bmatrix}\dot{x_1}\\\dot{x_2}\end{Bmatrix}=\begin{bmatrix}a_{11}& ...
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Particular integral equation

Let $a,\sigma, n>0$ be some parameters and define the conditional probability density function $$ p(x,y):= \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-n-x)^2}{2\sigma^2}\right). $$ Is it ...
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Calculating convolutions of probability density functions

I have a PDE: $$\frac{\partial N (x,u)}{\partial x}=\int _0^uN(x,u)f(u-u')du'$$ $$N(0,u) = \delta (u)$$ Here $f(u)$ is a probability density function for $0 \le u \le u_{max}$, $\int _0 ^ {u_{max}} ...
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Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) ...
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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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Heat equation via Laplace transform, final inversion…

Working on a PDE that may have solution out there somewhere but I have yet to stumble upon it. I prefer to use the Laplace Transform approach when I can, as I find it more intuitive than some other ...
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Transformed Laplace “solution space”

From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the ...
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Mellin transform and equality of two functions

If say, $\mathcal{M}f(s) = \mathcal{M}g(s)$ in a strip say $0 < a < \Re(s) < b$, then can we say that $f$ and $g$ are identical? Does the Mellin inversion theorem imply this?
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What polynomial transformation is this and how is it related to the original polynomial?

Let $f(X) = a_n X^n + \dots+a_0$. Then the Laplace transform of $f$ is $g(s) = \mathscr{L}\{f\}(s) = \frac{n! a_n}{s^{n+1}} + \dots + \frac{a_0}{s}$. If you now define the polynomial transform of ...
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Discrete to Continuous Representations of Functions via Laplace Transforms?

The Laplace transform can be thought of as the continuous analogue of a power series, as in this video. From this perspective, think of the function $ a : \mathbb{N} \rightarrow \mathbb{R}$ as a ...
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Laplace transform of a sum of stochastic variables

I have a problem with interpretation of one transformation performed on equation consisting of continuous random variables. Here is the source equation describing recurent relationship between the ...
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Inverse Laplace transform of functions with jump discontinuities

Given a function $F(s)$, suppose we define its inverse Laplace transform as: \begin{equation} f(t) = \lim_{k \to \infty} \frac{(-1)^{k}}{k!}\left(\frac{k}{t}\right)^{k+1}F^{(k)}\left( \frac{k}{t} ...
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Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...
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109 views

Final value theorem on coupled differential equations

Good day, I have two linear and coupled differential equations: $J_{11}\ddot{\theta_1}-n(J_{11}-J_{22}+J_{33})\dot{\theta_3}+n^2(J_{22}-J_{33})\theta_1=T_{c_1}+T_d \\ ...
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Laplace transform of $\sqrt{f(t)}$

I have a question about the Laplace transform. Suppose the Laplace transform of $f(t)$ is known. Is there any relation between the Laplace of transform $f(t)$ and that of $\sqrt{f(t)}$? Thanks in ...
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The Square of the Laplace Transform

I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find The norm of $\mathcal{L}^2$ The nullspace of The norm of $\mathcal{L}^2$ So ...