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

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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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The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
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Inverse Laplace Transform of $e^{c \cdot s^2}$

I am trying to find the Inverse Laplace Transform of the function $$ F(s)=e^{c \cdot s^2} $$ where $c > 0$.
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1answer
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Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
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1answer
23 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
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Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
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1answer
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How do you find the state space representation of $G(s) = \frac {1}{s^2+s+1}$

Let $G(s) = \frac {1}{s^2+s+1}$ be the transfer function of the system Then $Y(s)(s^2+s+1) = U(s)$ Therefore $y'' + y' + y = u$ After this step, how should I set up my state transition variable $x$ ...
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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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Solution of a heat transport PDE

Solve the system of partial differential equations: $$(1)\space\space \frac{\partial g}{\partial t} + v\frac{\partial g}{\partial x} = -k_1\left(g-h\right)$$ $$(2)\space\space \frac{\partial ...
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Example: How to find inverse Laplace Transform by integral of the function (5.2-29)

This is just a demonstration on how to solve the following type of problem. Find $\mathcal{L}^{-1}\{\frac{54}{s^3(s-3)}\}$ by the given method: $$\mathcal{L}\{ ...
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Prove that the Laplace transform of $I_0\left(2\sqrt t\right)$ 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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Laplace transform notation

I'm confused about the notation used, pretty much everywhere, to describe what a Laplace transform it. Wikipedia says something along the lines of "..Laplace transform of a function $f(t)$..", ...
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inverse laplace transform of F(s)

Let $f(x)$ be some arbitrary function , $F(s)$ is laplace transform of it I think inverse laplace transform of $\frac {F(s)}{s+r}$ where, r is constant may be $\int_0^t e^{-rt'}F(t') dt'$ ...
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1answer
27 views

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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1answer
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Inverse Laplace transform of $s^{\beta-1}/(s^{\beta}+a)$ [on hold]

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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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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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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1answer
29 views

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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1answer
19 views

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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539 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
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The inverse laplace transform of $p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$ can be written in Fresnel integrals?

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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What is the significance of laplace and fourier transform [closed]

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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Laplace transform of product of $\sinh(t)$ and $\cos(t)$

If I have a function $f(t)=\sinh(t)\cos(t)$ how would I go about finding the Laplace transform? I tried putting it into the integral defining Laplace transformation: $$ F(s)= \int_0^\infty ...
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Laplace transform of $y'' + 3y' + 2y = f(t), \; y(0) = y'(0) = 0$ [closed]

Can you do this? This is part of my final year EE work. I need to solve this in order to figure out how my sensor is behaving. $$y'' + 3y' + 2y = f(t), \; y(0) = y'(0) = 0$$ where $f(t)$ is a ...
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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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1answer
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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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Inverse Laplace transform of $\frac1{s-\lambda}$ is $e^{\lambda t}$

I got that the inverse Laplace of $\frac{1}{s-\lambda}$ is $e^{\lambda t}$. Does this look correct?
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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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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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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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How to justify this complex substitution using contour integration

I tried to solve the laplace transform of $\cos(at)$ and $\sin(at)$ using Euler's formula. That is, $$\int^\infty_0e^{-(s-ia)t}dt\color{red}{=}\frac{1}{s-ia}\int^\infty_0e^{-t}dt=\frac{1}{s-ia}$$ ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...