The Laplace transform is a widely used integral transform, similar to the Fourier transform.

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Inverse Laplace transform of $s^{\beta-1}/(s^{\beta}+a)$

I am stuck at calculating the inverse Laplace transform of $$\frac{s^{\beta-1}}{s^{\beta}+a}$$ where $0<\beta<1$ and $a>0$. Thanks
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Derive inverse Laplace Transform using two given trigonometric transforms (5.2-13)

I am not certain how to begin this problem. Someone please point me in the right direction. Problem Using the two given formulas ($1$ and $2$ below) show that: ...
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Why aren't my Laplace transform and Undetermind Coefficients answers matching up?

I might be losing my mind this morning (I am, for sure), but I can't these two techniques to give me the same answer to a basic differential equations problem. The problem is $y''-8y'+27y=0$ with the ...
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Methods of Solving Ordinary Differential Equations - A Small Question

I've spent some weeks now trying to learn how to solve ordinary differential equations, and I am now studying the Laplace transform and how this can be applied to solve ODEs. I feel a little bit ...
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Example: How to set up integral and find a Laplace Transform given two straigh lines (5.1-7).

This is an example on how to find the Laplace Transform for a graphical problem. The textbook solution we wish to derive is: $$F(s)=\frac{-e^{-s}}{s}+\frac{1-e^{-s}}{s^2}$$ We begin by expressing ...
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Computation of two-sided probability density functions from their cumulants using Laplace transform

The computation of one-sided probability density functions (PDFs) from their cumulants using Laplace transform is proposed by following paper: M.N. Berberan-Santos, Journal of Mathematical Chemistry, ...
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Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
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What is the significance of laplace and fourier transform [on hold]

I know what laplace and fourier transforms are used for but i want to know if these operators show some properties or are they just mathematical operators to simplify our work
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How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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yet another simple Laplace transform

what is $ℒ(t^2e^{3t})$ I have got this far so far: $=\int_{0}^\infty (t^2e^{t(3-s)})$ Integration by parts using: $u = t^2$ and $du = 2t$ $v = \frac{e^{t(3-2)}}{3-s}$ and $dv = e^{t(3-s)}$ Which ...
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23 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
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50 views

Laplace Transform of $e^{a t^2}$

What is the Laplace transform of $e^{a t^2}$, for positive $a$? In order for Laplace transform to exist function must be locally integrable. Since integral of any compact set $e^{a t^2}$ is finite ...
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1answer
12 views

Why is the Z-transform of $e^{at}$, t = kT, different from Laplace transform of $e^{at}$

The Laplace transform of $e^{at}$ takes a well known form of $\frac{1}{s-a}$ The Z transform of $e^{at} = e^{akT} $ T is the sampling period takes the form of $\frac{z}{z-e^{aT}}$ Does anyone know ...
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28 views

Laplace-Fourier transform issue

Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ we take the generalised Fourier transform $\hat{f}(w)=\int_{-\infty}^{+\infty}e^{iwx}f(x)dx$ where $w\in \mathbb{C}$. Now assume, this transform ...
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1answer
23 views

relationship between laplace transform and its derivative

By definition, the Laplace transform of a function $f$ is given by, $$ L(f)(\lambda) = \int_0^\infty e^{-\lambda s}f(s) ds .$$ My question is two fold. I need help in findding the derivative of ...
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28 views

What does (0+) mean?

I'm currently learning from a script (which is written in German and not publicly available, sorry) for introduction to stochastics, where the topic is the Laplace transformed function for random ...
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1answer
18 views

Prove that the Laplace transform of $L_t\left[I_0\left(2\sqrt t\right)\right]$ is $\exp\left(1/s\right)/s$

Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it ...
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24 views

regarding the integral form of the inverse Laplace transform

The integral form of the inverse Laplace transform is given as $$f(t)=\frac{1}{2\pi i}\int_{s'-i\infty}^{s'+i\infty}e^{st}F(s)ds$$ where $s'$ is larger than the real parts of all the possible ...
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2answers
48 views

Beginner question about existence of Laplace transform

I am having problems understanding why a Laplace transform exists or not. Here is my math and logic, hopefully someone can point out where I am wrong. $$f(t)=e^{at} \implies ℒ[e^{at}] = ...
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17 views

Laplace transform to minus infinity

This is a solution to $f(t) = \Theta(t) - 1$ found in my textbook. $$ Lf(s) = \int^{+\infty}_{-\infty} e^{-st}(\Theta(t) - 1) dt = \int^0_{-\infty} e^{-st} (-1) dt = \left[ \frac{e^{-st}}{s} ...
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36 views

Find the laplace transform of $u(-t+a)$, u is the step function

I am not sure how to deal with the minus sign in front of t But we can try: $$U(s) = \int_0^\infty u(-(t-a)) e^{-st}dt$$ which leads to $$U(s) = \int_{0}^a e^{-st}dt$$ Is this correct?
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How to intuitively understand why Laplace transform has a planar region of convergence, where Z transform has a circular region of covergence

Is there a more profound insight that can be seen going from the Z-transform: $X(z) = \sum_{n=0}^{\infty}x[n]z^{-n}$ To the Laplace transform: $ F(s) = \int_0^\infty f(x)e^{-sx}~dx$ That allows ...
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Borel-/Laplace-transform and $\psi$-function

I'm considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A (of course thought as of infinite size) $ \qquad $ The ...
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36 views

Good recommendations for solving PDE's by integral transforms

I look for good books on solving partial diffrential equations (PDE's) using integral transforms specially Fourier and laplace transforms. Do you have any recommendations for such books? I don't ...
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22 views

laplace transformation solve heaviside d.e. $y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$

$y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$ I did the transformation and obtained $Y=e^{3s}(\frac{1}{s^2}-\frac{2}{s}-\frac{1}{s^2}+\frac{2}{s+1})+(\frac{3}{(s+1)^2}+\frac{2}{(s+1)})$ This ...
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61 views

Solving a pde with laplace transforms

Consider the following problem: $$∂g/∂t − ∂/∂x(x^{2−a} ∂g/∂x) = δ(x − ξ)δ(t − τ )$$ with $0 < x, ξ < ∞, 0 < t, τ,$ where the solution remans finite over the entire interval and initially ...
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41 views

laplace transformation solve heaviside d.e. $y''+4y=U(t-4)$

$y''+4y=U(t-4)$ so that $y(0)=3$ and $y'(0)=-2$ I have applied the transformation in both terms obtaining $Y=\frac{3s^2+10s+1-e^{4s}}{s(s+4)}$. How can i solve it?
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use laplace transformation to solve $y^{iv}-16y=0$, being $y(0)=1$, $y'(0)=0$, $y''(0)=0$, $y'''(0)=0$

Folowing the process, i came to $Y=\frac{s^3}{s^4-16}$ However, when trying to write the fraction as a sum of other fractions,the system is undetermined. ...
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Solving a differential equation using Laplace transform?

$$y''+2y'+ 10 = b\,δ(t-T),\,\begin{cases}y(0)=3\\ y'(0) = 0\end{cases}$$ I managed to solve this equation. My answer is $$y(t) = 3e^{-t} \cos(3t) - ...
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Find what values of 'b' have bounded solution(differential equation)?

$y′′ + b^2{y} = f(t)$ $ f(t) = t$ for $0 < t < 2\pi$ ($2\pi$ periodic sawtooth wave) This is my solution to the differential equation. $y(t) = C_1\cos(bt) + C_2\sin(bt) + b^{-3}\left(bt - ...
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Use Laplace transformations to solve $y''+4y'+4y=e^{-x}$, so that $y(0)=0$ and $y'(0)=1$

Use Laplace transformations to solve $y''+4y'+4y=e^{-x}$, so that $y(0)=0$ and $y'(0)=1$. I applied the transformation but I don't understand the rest of the process. Can anyone explain me based on ...
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inverse laplace transformation of $\arctan(\frac{4}{s})$

inverse laplace transformation of $\arctan(\frac{4}{s})$ using I was trying use 12 but i couldn't arrive to a solution
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laplace transformation $\cos^2(3t)$ and $\sin(5t)cos(2t)$

it is asked to transform $\cos^2(3t)$ and $\sin(5t)cos(2t)$ using the results from i think the process might be similar for both of them but i don't know wich result to use. can you help me? ...
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Laplace tranform of $t^{5/2}$

It is asked to transform $t^{5/2}$. I did $t^{5/2}=t^3\cdot t^{-1/2}$. Then followed the table result $$L\{{t^nf(t)}\}=(-1)^n\cdot\frac{d^n}{ds^n}F(s)$$ However i got $\frac{1}{2} \cdot\sqrt\pi ...
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54 views

Equality of laplace transform

Assuming that Laplace Transforms of two functions $f$ and $g$ are equal, is it true that $f=g$? There is one-to-one correspondence between functions and their Laplace Transforms, so it seems to me ...
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Laplace transform: Convolution

Let $H(s) = \frac{1}{(s^2 + w^2)^2}$ Then $\displaystyle h(t) = \frac{\sin(wt)}w * \frac{\sin(wt)}w = \frac 1{w^2} \int_0^t \sin(w \tau) \sin(w(t-\tau)) \,d\tau$ $\displaystyle = \frac 1{2w^2} ...
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19 views

Abscissa of Convergence for the Laplace Transform of $f(t)=e^t \sin(e^t)$

I am trying to solve the following question: Show that the abscissa of convergence for the function $f(t)=e^t \sin(e^t)$ is zero, i.e the unique number $\sigma$ such that the integral $\int_0^\infty ...
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How to find inverse laplace transform of $\frac{2\sqrt s}{2\sqrt s+1}$

How to find inverse laplace transform of $$\dfrac{2\sqrt s}{2\sqrt s+1}$$ I tried to solve it, but I couldn't.
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How to solve IVP using Laplace transform(of matrix)?

$$x' = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & -2 \\ 3&2 & 1\end{bmatrix} x, ~~ x{(0)} = \begin{bmatrix} 2 \\ -1 \\ 1\end{bmatrix}$$ I am having very hard time solving this ...
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finding the inverse Laplace transform of $\frac{1}{z\sqrt{z+1}}$

i know that the inverse Laplace transform is given by $$2\pi i \left\{\sum\space\text{ of the residues at the poles of}\space e^{zt}f(z)\right\}- \frac{1}{2 \pi i}\int \text{ along the branch cut}$$ ...
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Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the Laplace transform of the function $\displaystyle f(t)=te^{-t}\sin(2t)$ using only the properties of Laplace transform, meaning, use clever tricks and the table shown at ...
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38 views

Pollaczek-Khinchin formula for ruin probability - proof

I got stuck in a specific part of proof of the Pollaczek-Khinchin formula (in book "Stochastic Processes for Insurance and Finance", T. Rolski et al., section 5.3.3, theorem 5.3.4). Namely, why the ...
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Check my answer - simple laplace transform of piecewise continuous function.

I'd just like to check that I got the idea right, first exercise im doing in laplace transforms and am a bit clueless. We are given $f(t)=0$ if $0<t<2$ and $f(t)=t$ if $t>2$. We are asked to ...
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39 views

Laplace Transform of the square of first derivative

I want to compute the laplace transformation of this equation: $$4 -0.1{\operatorname{d}\!x(t)\over\operatorname{d}\!t} - 0.01\left({\operatorname{d}\!x(t)\over\operatorname{d}\!t}\right)^2 = ...
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31 views

Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
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Inverse Laplace of $\frac 1 {(s^2+a^2)^n}$

How to compute the Inverse Laplace of $\frac 1 {(s^2+a^2)^n}$? I know that to compute Inverse Laplace $\frac 1 {(s^2+a^2)^2}$, the convolution Theorem is useful. but is there an interesting idea for ...
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Solution of given differential equation using Laplace Transforms.

I need solution of DE $$y'' + 2y' + 5y = 0$$with initial conditions $$y(0)= 1 \text{ and } y'(0)=0$$ I tried this but problem came when i started taking laplace inverse of F(s), so i need a complete ...
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1answer
21 views

Inverse Laplace Transformation

I was solving a problem but I am stuck at it. Here is the question : $\frac{7s^2+9s+3}{(s^2-12s+40)(s^2+9)}$ Find inverse Laplace transform. I performed these operation : ...
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1answer
34 views

Inverse Laplace Transformation

I have a question about laplace transformation. $\frac{8s+4}{s^2+23}$ I tried to split them. $\frac{8s}{s^2+23}$ is the image of a cosine and $\frac{4}{s^2+23}$ is the image of a sine. Here is ...
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Transforming Exponential to Ordinary Generating Functions

I am looking for a particular ordinary generating function, if it exists for the Associated Stirling Numbers of the second kind $$b(1;n,j)=b(n,j)=\sum_{k=0}^j(-1)^k\binom{n}{k}S(n-k,j-k)$$ Where ...