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

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Does the scaling property of Laplace transforms also hold for complex scaling?

Consider the following Laplace transform (it arose in the context of Borel resummation) $$ \int_0^{\infty}e^{-\zeta}\phi(z\zeta)d\zeta $$ my textbook says that $$ \int_0^{\infty}e^{-\zeta}\phi(z\zeta)...
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Initial Value Problem using Laplace transformation: What is the ${\cal L}$ transform of $u(t-5)$?

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses a single difficulty. If it's any indication of difficulty, the exercise ...
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25 views

Asymptotic behavior of inverse laplace transform [duplicate]

My question may be quite rough. Let $F(\lambda)$ be the Laplace transform of some function $f(t)$, $$ F(\lambda)= \int_0^\infty e^{-\lambda t}f(t) dt. $$ If I have knowledge about $F(\lambda)=O(\...
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Statistics - Laplacian Characteristic Function

I was asked to find the characteristic function of the Laplacian random variable. And, from that, to find the mean of it. Im having trouble solving this. That's what I have done so far, but it doesn'...
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69 views

How to calculate the inverse Laplace transform of $F(s)=\frac{1}{1-e^{-s}}$?

How to calculate the inverse Laplace transform of this function? $$F(s)=\frac{1}{1-e^{-s}}$$
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Trigonometric functions and complex numbers

I solving the inverse Laplace transform using the method of Heaviside. This is part of the problem: I understand the division between complex numbers and that $e^{it} = Cos(t) + iSin(t)$, but I ...
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50 views

How to solve a differential equation with a distributional free term?

I tried to solve this type of differential equation $$y'' + y = \delta + \delta' .$$ I tried using the Laplace Transform, but I'm stuck at that $\delta$ (Dirac function). The only thing I know is ...
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20 views

What exactly is the variable of Laplace transform

I try to find the solution for a hard differential equation. I could not solve it with any orthodox method. However, if I use Laplace transform and then replace its term with its Maclaurin series, it ...
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Laplace and Transfer Function Problem

I have to show that all initial condition problem where $t = 0$ with constant coefficients: $ a_{n} y^{n}(t) + a_{n-1}y^{n-1}(t)+...+a_{2}y''(t) + a_{1}y'(t) + a_{0}y(t) = x(t) $ with $ y^{k} = y_{...
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extracting a function out of an equation

I encountered the following problem in my thesis. We have an equation as follows: $\phi(s)=\sum^\infty_{n=1}P(n)\int^{\infty}_{0}e^{-st}f(t|n)dt=\sum^{\infty}_{n=1}[(1-q)M_1(s)]^{n-1}qM_2(s)$ in ...
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Solve non-linear pde

so i wonder the next thing : if i consider a pde like heat equation Fourier transform works very well. Now, if i consider this equation : $\frac{\partial u(t,x)}{\partial t}-k(u(t,x))\frac{\partial^2 ...
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57 views

What is the inverse Laplace transform of $\lfloor s \rfloor$?

How can we find the inverse Laplace transform of: $[x]$ (floor function) ? My question isn't LLaplace transform of floor function i asked the "inverse" laplace transform of floor function $$\mathcal{...
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1answer
17 views

How do I proove that function is solution of the Laplace equation?

How do I proove that for $\vec{r}=(x,y,z)\in \mathbb{R}^3,\vec{r}\neq 0$, function is $u(x,y,z):=1/(-ln\left \| \vec{r} \right \|)$ a solution of the Laplace equation $\Delta u=\frac{\partial^2 u}{\...
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1answer
26 views

Laplace Transform of $t^2$ , for $t\ge1$ .

$$f(t) = t^2 , t>=1$$ $$f(t) = 0, 0<t<1$$ what is the laplace transform of $f(t)$, It is solved In my sheet as $$t^2 = (t-1)^2 + 2t -1 $$ $$ L(f(t)) = e^{-s} L(t^2 +2t +1) = e^{-s}(2/s^3 + ...
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Laplace transform to solve a pde

i have to find the laplace transform of this : $$f(u(x,t))\frac{\partial u(x,t)}{\partial t}$$ I have no idea how to solve this... Thank you for your help.
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27 views

Laplace transform using change of scale property [closed]

If Laplace transform of $f(t)=\phi(s)$, then Laplace transform of $e^{bt}f(at)$ is
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47 views

prove the result of a Laplace transformation

I have to prove the next problem $$\mathcal{L} \left(\int_{0}^{t}\frac{1-e^{-u}}{u}du,s\right) = \frac{1}{s}\log\left(1+\frac{1}{s}\right)$$ I'm quite new in the subject and I have troubles with ...
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14 views

Laplace Transform on this Poisson Process?

Two independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$. The waiting is exponentially distributed with mean of $\frac{1}{\mu_1}$ and $\frac{1}{\mu_2}$. Knowing this, how to ...
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41 views

Why is Laplace Transform of $\delta(t)$ $F(s)=1$, not $0$ or $\frac{1}{2}$?

Let $\varepsilon \in \mathbb{R}$, the following integral $$ \int_{\varepsilon}^\infty \delta(t)e^{-st}dt. $$ converges to $1$ if $\varepsilon \to -0$ and $0$ if $\varepsilon \to +0$. This shows ...
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20 views

laplace transform probability distribution not concentrated on 0

This seems intuitively obvious but how to prove that $\hat{\mu} < 1,$ when $\theta >0$ and $\mu$ is a probability measure not concentrated at $0,$ where $\hat{\mu}$ is defined as below $$\hat{\...
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1answer
13 views

Get transfer function of a nonlinear diff. equation

I have this equation: $$\frac{\partial v}{\partial t} = -g + c\left(u(t) - v(t)\right)^2$$ g and c are constants. u(t) is my input and v(t) is my output. I need to reach the transfer function $\frac{...
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37 views

Non-trivial inverse Laplace transform

I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral $$ F(t) = \frac{1}{2 \pi i} \int_{- ...
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Laplace transform of two exponential distribution

1) Type 1 clients take exponential time with average of $\frac{1}{\mu_1}$ to be served. 2) Type 2 clients take exponential time with average of $\frac{1}{\mu_2}$ to be served. 3) Type 1 and 2 ...
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2answers
23 views

Alternative integration limits in a Laplace transform

The unilateral Laplace transform of $f(t)$ is $\int_0^\infty e^{st} f(t) \mathrm{d}t$. If we define the transform as $\int_{a}^\infty e^{st} f(t) \mathrm{d}t$, would it conserve all the nice ...
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40 views

Workbook recommendation in preparation for Electrical Engineering

I'm currently preparing myself for starting my graduate degree in Electrical Engineering. The mathematics courses given are outlined as follows: Mathematics 1 Real functions Continuity, limits, ...
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1answer
23 views

Laplace Transform - Partial Fractions [closed]

Please see attached image. I keep coming up with a irrational/complex coeffecient which is correct. Can you please help me put it partial fractions please Thanks
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1answer
23 views

Inverse laplace transform using equating coeffecients method [closed]

Please see attached image. Could you please help me do the inverse of this laplace transfrom. I'm using the method of trying to equate the coeffecients. Thanks
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1answer
68 views

Coupled differential equations: Laplace transform

By using Laplace transforms find the steady state values of w, u and v:$$\frac{du}{dt}=-\frac{\Gamma}{2}u+\Delta v,$$$$\frac{dv}{dt}=-\frac{\Gamma}{2}v - \Delta u +w \Omega,$$$$\frac{dw}{dt}=-\Gamma-\...
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60 views

Solving ODE using Laplace transformation without table of transform

Solve $$y'' + 4y' -5y = 2xe^{-3x}$$ using the Laplace transform method. For the the Laplace transform of the ODE you must do the integration (do not use table). Table can be only used when computing ...
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28 views

Laplace of a function raised to a power

For example: $y' = y + y^2$ The Laplace of the first two terms is $s(F(s)-f(0))$ and $F(s)$. But what is the Laplace of $y^2$?
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57 views

Differential equations using Laplace transforms

By using Laplace transforms find $x(t)$ from the coupled differential equations$$\frac{dx}{dt} = -kx+gy+E,$$$$\frac{dy}{dt} = -ky-gx,$$for some functions $x(t),$ and $y(t)$, where $E, k, g$ are real. ...
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26 views

Laplace transform of a convolution-like function

Is there a way to calculate the Laplace transform of the following function? $$ \sum_{k=1}^{+\infty}f(t-(g(t)-\theta_k))h(g(t)-\theta_k), \qquad t>0. $$ Thanks in advance.
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Consider the equation $y'' + 4y = f(t), y(0) = 1, y'(0) = 0$. Use the Laplace Transform to compute the Green’s function for this equation.

Consider the equation $y'' + 4y = f(t), y(0) = 1, y'(0) = 0$. Use the Laplace Transform to compute the Green’s function for this equation. $y'' + 4y = f(t) \rightarrow L\{y'' + 4y = f(t)\}=L\{f(t)\}$...
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Laplace transform problem [closed]

This is the problem. I know ℒ (t^n) = ℒ (t^n-1) x n/s, which should give us -2/sqrt(s). That's as far as I've come and I don't get the hint either. Where does the pi even come from?
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Cross-correlation, Fourier transform and Laplace transform: measure of how much signal are alike?

I'm studying electrical engineering and use correlation, Fourier transform and Laplace transform a lot. I know how and when to use them, however, the interpretation I've seen in the lectures still ...
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43 views

Solving differential equation using Laplace transform, problem finding inverse

Given $$y'' + 4y' + 5y = H(t-3)e^{-2t}, t>0, y(0) = 1, y'(0)=2 $$ To solve this diff. equation using Laplace transform. Seems very straightforward. On one side, we have $$\mathscr{L}\{y''+4y'+5y\} ...
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1answer
26 views

Sine Curve Circular Transform - Parametric Equations

Is there a way to transform a sine curve so that the x-axis of the sine curve would become a circle, with the sine wave oscillating around the now-circular x-axis? What would be the parametric ...
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33 views

What is the domain and codomain of a transfer function? [closed]

Let's say I have the transfer function- $\textbf{H}(j\omega)=\cfrac{1}{1+j\omega RC}$ Where does this function map to and from, and can it be plotted visually?
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1answer
41 views

Finding the inverse Laplace transform of this function

Find the inverse Laplace transform of this function (related to my question earlier): $$f(t)=\mathcal{L}_s^{-1}\left[\frac{s}{s+\frac{1}{\tau}}\cdot\frac{A}{s}\left(1-\mathrm e^{-\frac{Ts}{2}}\right)^...
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2answers
60 views

Circuit Analysis problem (find the problem)

In this question, I know that $\text{C},\text{R},\text{T},\text{A}\in\mathbb{R}^+$ I've this circuit (the bottom of the resitor is connected to earth ($0$)): When I use Laplace transform I can find ...
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1answer
21 views

Expressing the Weierstrass transform in terms of the unilateral Laplace transform

I was looking for a way to express the Weiestrass transform of $f(t)$, $$\mathcal{W}\{f(t)\}(s)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(t)\exp\left(-\frac{(s-t)^2}{4}\right)dt$$ in terms of the ...
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1answer
23 views

Laplace transform ODE

Use laplace transform to solve the ODE $y''(t)+4y(t)=4u(\pi-t)cos(t)$,,,,,,, $y(0)=y'(0)=0$ u is the unit step function (heaviside function) I use: $u(\pi-t)=1-u(t-pi)$ By inserting this and ...
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Solve $y'(t)=t+1$ using Laplace transform

I'm studying how to use Laplace transform to solve ODEs. I have thought to use this very simple example: $$y'(t)=t+1 \qquad y(0)=0$$ I can use integration to find $y(t)$: $$y(t)=\int (t+1) \ \ dt=\...
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1answer
32 views

Laplace transform of a square wave function

What is the right way to find the Laplace transform of this function: The thing I noticed was: $$f(t)=\text{A}\space\space\space\space\space\space\space\space\space\space 0\le t<\frac{\...
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1answer
44 views

Invere Laplace transform of a function (related to circuit analysis)

I'm studying circuit analysis. I've to solve this inverse Laplace transform to see the response: $$\mathcal{L}_{s}^{-1}\left[\frac{r}{r+\frac{1}{cs}}\cdot\frac{k\tanh\left(\frac{as}{2}\right)}{s}\...
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28 views

Using laplace trasnform find the convulation of (f*g)(t).

$let \;f(t)=\sin (3t)$ and $g(t)=e^{-2t}$ Using laplace trasnform find the convulation of (f*g)(t). convulation theorem: $h(t)=(f*g)(t)= \int ^t_0 f(u) g(t-u) du$ $L(sin (3t)=\frac{3}{s^2+9}\\ L(e^...
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25 views

Inverse Laplace of an exponential function $\exp{\{-x\sqrt{(s+h)/k}\}}$

I am having difficulty to figure out to use a Laplace Transform Table formula to verify a particular case. The inverse Laplace transform of $$L^{-1}\bigg[\exp{\bigg\{-x\sqrt{(s+h)/k}\bigg\}}\bigg]$$ ...
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The integral $\int_{o}^{\infty} e^{-3t} cos(4(t-5))u(t-5) dt$ using laplace transform

$$\int_{o}^{\infty} e^{-3t} cos(4(t-5))u(t-5) dt$$ I need to find the laplace transform. Do i consider it as a convolution integral or as f(t)/t and work accordingly. Thank you. Edit: It turns out ...
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First order IVP with unknown continuous function

Solving $y''+4y=g(t)$, $y(0)=3$, $y'(0)=-1$ using Laplace Transforms. I get $Y(s)=\frac{G(s)}{s^{2}+4}+\frac{3s}{s^{2}+4}-\frac{1}{s^2+4}$ Then using Inverse Laplace Transforms, I am not sure of my ...
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29 views

Stuck relating the input to output (Transfer function)

i want to find the transfer function of a differential equation (given below) $\ddot\theta = a [ ([b\times Xin] - bk\dot\theta) - \ddot\theta] - c\phi $ (where $\phi$ and $\theta$ are time ...