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

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Laplace transform of the square of Brownian motion hitting time

Let $B_{\mu}(t)$ be a one-dimensional Brownian motion with drift $\mu \geq 0.$ For $a > 0,$ let $$T_a = \inf\{t > 0: B_{\mu}(t) = a\}$$ denote the first hitting time of $B.$ The Laplace ...
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Finding the inverse laplace of this function: $ F(s)= \frac{s+8}{s^{2}+4s+5}$

Im trying to find the inverse laplace of : $ F(s)= \frac{s+8}{s^{2}+4s+5}$ I reached the following: $$ F(s)= \frac{s}{(s+2)^{2}+1} + 8 \times \frac{1}{(s+2)^{2}+1}$$ Now i have the 2nd term in the ...
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33 views

Inverse Laplace Transform by Partial Fraction Expansion

I've been trying to solve this partial fraction for a Laplace transformation but I can't. Is there any way to solve it? $$\frac{(s-t)^2}{((s-t)^2-1)((s+1)^2+4)}$$ Could somebody help, I've been ...
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inverse laplace transform of $\frac{s^3}{(s^2+4)^2}$

Using partial fractions gives $\frac{s}{s^2+4}$ - $\frac{4s}{(s^2+4)^2}$ Inverse laplace transform of the first member ($\frac{s}{s^2+4}$) is cos(2t). Can't figure out how to transform $\frac{4s}{(s^...
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What is the Inverse Laplace of $1\over (s^2-a)$.

What is the inverse Laplace of $1\over (s^2-a)$. Since $L(sinh(at))={a\over s^2-a^2}$ can I take the inverse Laplace of $1\over (s^2-a)$ as, $sinh(\sqrt at)\over \sqrt a$
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34 views

Taking the inverse Laplace under boundary conditions

I want to solve the problem as described here in this article I am trying to solve it using Laplace Transformation. This is what I did: Taking Laplace transformation of the equation I get $$K[s^2\...
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73 views

Laplace functional of sum of independent uniformly distributed random variables

I'm doing some of the exercises in Cinlar's "Probability and Stochastics" to better understand the material. This exercise (VI.1.17) is taken from page 247: Fix an integer $n \geq 1$. Let $X_1,\...
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Problem in finding Inverse Laplace

I have the following equation But if I let $T_t(x,t)=constant$ then my equation becomes at steady state, since partial differentiation of a constant=0, right? here $\omega_bp_bc_b=M, $a ...
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46 views

Laplace transform to bio heat equation

This is the bio heat equation and I have several questions when trying to work with it. $$ \rho c \frac{\partial u(x,t)}{\partial t} = \nabla[k \nabla u(x,t)] + \omega_b \rho_b c_b [u_a - u(x,t)] ...
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Laplace inverse of $\frac{e^{-\sqrt{s+2}}}{s}$

I want to find out $$\mathcal{L^{-1}}\{\frac{e^{-\sqrt{s+2}}}{s}\}$$ How do you find the inverse Laplace? thanks
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Deriving the Laplace transform of $t$

In deriving the Laplace transform of $t$, I'm not being able to grasp why the limit term go to 0: $$\begin{align} x(t)&=t\\ \\ \therefore\; X(s)&=\int_0^\infty t\cdot e^{-st}\;\mathrm{d}t\\ &...
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31 views

Solving a differential equation that includes cosine

Anyone interested in coming up with a concise equation for $u(\tau)$ given the equation for its derivative below? \begin{align} \frac{du}{d\tau}=-\sigma u + S\bigg(1+B\cos(\tau)\bigg) \end{align}
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Find the coefficient of partial expansion $\frac{2x+3}{(1-x)(1+0.5x+0.5x^2)}$

I want to decompose the equation: $$\frac{2x+3}{(1-x)(1+0.5x+0.5x^2)}=\frac{A}{1-x}+\frac{B}{1+0.5x+0.5x^2}$$ I found $A$ by multiple both side with $1-x$ and plug $x=1$. However, it is so difficult ...
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207 views

Moment Generating Function and Inverse Laplace transform

I need to compute the inverse Laplace transform of the function $$ M(t)=e^{\frac{t^2}{2}} $$ Now, I know that this is a normal distribution with mean zero and variance 1, but how the computations are ...
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Attempting to use the Laplace transform to solve a second order ordinary differential equation with a piece wise forcing function.

Thanks to everyone who will bare with me and read this attempt and further validate it correct and improve upon it or otherwise correct its wrong. The question: $$ y''(t)+2y'(t)+y(t)=u\left( t-\...
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27 views

For which $s$ is the Laplace transform of Dirac delta function defined?

I have to find for which $s$ the Laplace transform of the Dirac delta distribution is defined. My idea is to start with the definition of the Laplace transform: integral from 0 to infinity. However, ...
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24 views

Show $\int_{s}^{\infty} f(x)dx = \mathcal{L} \{\frac{F(t)}{t}\}$ given $f(x) = \int_{0}^{\infty} e^{-xt}F(t)dt$

I'm trying to derive this to show that $$\int_{0}^{\infty} f(x)dx = \int_{0}^{\infty} \frac{F(t)}{t} dt$$ and use that to prove $$\int_{0}^{\infty} \frac{\sin t}{t} = \frac{\pi}{2}$$ How do I go ...
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Inverse Fourier transform using laplace

We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they ...
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98 views

Bilateral Laplace transform

My knowledge of Bilateral Laplace transform is less. Here are the few questions I need answer. What is the condition for existence of bilateral Laplace transform? How is the condition for existence ...
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108 views

Show that $Y(s)$ satisfies $(1+s^2)Y'(s) + sY(s) = 0$ for $ty'' + y' + ty = 0$

I first approached the problem by finding the Laplace transform of $ty'' + y' + ty = 0$, such that: $ts^2Y(s) - tsy(0) - ty'(0) + sY(s) - y(0) + tY(s) = 0. $ I solved for $Y(s)$: $Y(s) = \frac{tsy(...
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laplace transform of difference between two gamma independent random variables

Knowing that the laplace transform of a Gamma distribution is given by: $$F_x(s) = \frac{\beta^a}{(s + \beta)^a}$$ and that for Z = X + Y "Sum of two independent Gamma distribution random variables"...
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If $\int_{0}^{\infty} f(x) \, dx $ converges, will $\int_{0}^{\infty}e^{-sx} f(x) \, dx$ always converge uniformly on $[0, \infty)$?

I previously asked about sufficient conditions to conclude that $$\lim_{s \to 0^{+}}\int_{0}^{\infty} e^{-sx} f(x) \, dx = \int_{0}^{\infty} f(x) \, dx$$ when $\int_{0}^{\infty} f(x) \, dx$ does not ...
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27 views

What is the inverse Laplace transform of $\frac{p^2}{(p^2+4)^2}$

Given $$f(p)=\dfrac{p^2}{(p^2+4)^2}$$ So $$f(p)=\dfrac{p^2}{(p^2+4)^2}=\dfrac{p^2+4-4}{(p^2+4)^2}=\frac{1}{p^2+4}-\frac{4}{(p^2+4)^2}$$ I know the inverse Laplace transform of the first term but ...
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48 views

Convolution of exponential and rect functions

I have a convolution question in my signals and systems problem set that is puzzling me: $ f(t) = e^{-t/2T} u(t) $ and $ g(t) = rect(t/2T) $ find the convolution $f \ast g$ and I am assuming $T>0$...
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Questions on Inverting Laplace transforms and Probability

From Williams' Probability w/ Martingales: Are we allowed to switch derivative and integral as follows: $$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = \int_{0}^{\...
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Laplace Transform of the Unit Step Function

Finding the Laplace transform of: $$f(t) = \begin{cases}1 ,& 0 \leq t < 1 \\t^2 ,& 1 \leq t \leq 2 \\ 4, & t \ge 2\end{cases}$$ Heaviside /Unit step function: $$1+ (t^2 -1)u(t-1) + (4-...
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Laplace transform with Heaviside Unit step function

Working on a question and was wondering if this is right. Find the Laplace transform: $$x'' + 2x' + x = f(t)$$ $$x(0)=x'(0)=0$$ $$f(t) = \begin{cases}2 ,& 0 \leq t \leq 2 \\t ,& t \ge 2\end{...
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How do I Inverse Laplace $\frac{(s+1)^3}{s^4}$

I missed a class this week in maths and been a bit lost since with Inverse Laplace, how do I go about finding the Inverse laplace of: $$\frac{(s+1)^3}{s^4}$$ Do I simply expand the numerator? then ...
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Analytic solution of the equation $c\int_0^t x^{a-1}e^{-x}dx + (c+e^t)e^{-t}t^{a-1} = 0$

I would like to find the closed form solution of the equation in the title for the parameter $t$ when $-1<c<0$ and $0<a<1$. I tried to use the Laplace transform. The transformation of ...
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Understand first step of Laplace transform of integral

I am new to Laplace transform, and have some hard time understanding the very first step of the "preparation" before taking the laplace transform. $${f(t) =\int_0^t u \cosh(3u)\,\mathrm{d}u } =\...
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The Laplace transform of an integrable function is differentiable

let $f\in L^1(0,\infty)$. For x>0, define $g(x)=\int_{0}^{\infty} f(t) e^{-tx} dt$. Prove that $f$ is differentiable for $ x>0$ and with derivative $g'(x) = \int_{0}^{\infty} -tf(t) e^{-tx} dt$. ...
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System response (TF), multiplication vs substition

Let's say I have modeled a system as a transfer function: $H(s) = \frac{Y(s)}{U(s)}$ Given the question: For which values of $a$ is the input signal $e^{a t}$ absorbed by the system $H$? What is ...
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Stopping time distribution and transforms with 1-dimension B-motion.

Let $W_t$ be a 1-dimensional Brownian Motion. For $x>0$, we define: $$\tau_{x} = inf \{ t \geq 0; |W_t| = x\}$$ Compute $E[e^{-s\tau_x}]$ and prove that $\tau_x$ is equal in distribution to $x^2\...
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$\mathcal{L}^2$-norm of the Laplace transform

I have been considering the Laplace transform $$\mathcal{L}(f)(s)=\int_{0}^{\infty}{f(t)\, e^{-st}dt}$$ defined on $s\in\mathbb{R}^{+}$ as an linear operator from $\mathcal{L}^{2}(\mathbb{R}^{+},\...
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differentiation

I wanna know how can I get Laplace transform by differentiation of this unit step function $$f(t)= u(t)-u(t-k)+2(u(t-k)-u(t-2k))+3(u(t-2k)-u(t-3k))+\ldots$$ I found that $$F(s) = \frac1s(1+e^{-ks}...
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Integral parameter equation

How to solve the following equation, $$\int_{-i\infty}^{+i\infty} e^{st}\frac{\Sigma_{k=0}^{m}b_ks^k}{\Sigma_{k=0}^{n}a_ks^k} ds=0$$ to obtain $t$ where, $m\le n$ and $a_i$, $b_i$ are the given ...
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Finding the maximum of step response for a given transfer funtion

Assuming that the following transfer function is given: $$F(s)=\frac{\Sigma_{k=0}^m b_k s^k}{\Sigma_{k=0}^n a_k s^k}$$ $$m\le n$$ Lets say $g(t)$ is the step response to this transfer function. ...
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Use Laplace transform to solve the system

$\frac{\partial x}{\partial t} = -y$ , $\frac{\partial y}{\partial t} = x.$ For $t\geq 0$ , with $x(0) = 1$, $y(0) = 0$. I have no clue on how to start the question. Could some please give me a ...
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Laplace transform and distributions

I was studying for my course in Fourier Analysis and was going through some old exams, when this question came up: Let $s^{-1}_+$ and $s^{-1}_-$ denote the analytic distributions given by the ...
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Solving IVPs with Heaviside function where switch depends on function value

I am looking to solve an IVP of the format $y'(t) = \begin{cases} a, & y(t) < y_0 \\ b, & y_0 \le y(t) < y_1 \\ c, & y(t) \ge y_1 \\ \end{cases}$ That ...
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Can the z-transform/laplace transform be generalized?

The fourier transform is a special case of z/laplace where the countour being projected on is the unit circle. The ztransform/laplace transform then generalizes this to a circle of any radius. Is ...
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Laplace Transform of $\int_{0}^{t}u^2e^u\,du$

Good day, I am having some problems with the Laplace transform of $f(t) = \int_{0}^t{u^2e^u\textrm{d}u}$. $$\mathcal{L}\left [\int_{0}^t{u^2e^u\textrm{d}u} \right ]$$ using the property : $$\...
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Inverse Laplace transform of the form $F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$

I am trying to solve the inverse Laplace transform of the form \begin{equation} F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}} \end{equation} where, $a$ and $b$ are known constants, $m$, $n$, ...
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understanding basic Laplace transformation

I am trying to understand Laplace transformations. Could someone tell me from where the fraction (1/-s), in red on the first line is originated? http://imgur.com/bTqjLGl
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Inverse Laplace transform of a hypergeometric function

I managed to solve an initial value problem in the Laplace domain in terms of a special function $ F(s) = c_2 \frac{1}{{{\left( {{s}^{1 +\beta}}-1\right) }^{\frac{1}{\beta+1}}}}+ c_1 \frac{s}{{{\left(...
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26 views

How do you differentiate a Laplace transform?

Consider the Laplace transform of $\color{green}{t\cfrac{\mathrm{d^2}f}{\mathrm{d}t^2}}$: $$\mathcal{L}\left[{t\cfrac{\mathrm{d^2}f}{\mathrm{d}t^2}}\right]=\int_{t=0}^{\infty}e^{-st}{t\cfrac{\mathrm{...
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15 views

Find maximum $p$ so random variable is in $L^p$ using Laplace transform

For a (non-negative) random variable $X,$ define its Laplace transform as $$\mathbb{E}(e^{-\lambda X}) =: L(\lambda),$$ for $\lambda > 0.$ We know that this uniquely determines the law of $X$ as ...
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existence of laplace transfrom to 1

I'm asked to find existence of f(t) given equation in exam 1 * f(t) = 1 (* means convolution) I tried to solve it by transforming (1/s) x F(s) = (1/s) so I got F(s) = 1 and then I said f(t) is ...
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13 views

Solving $\frac{dV}{dt} = \frac{I}{C}$ using the Laplace transform

I have the following equation for the evolution of the membrane potential ($V$) of a neuron: $$ \frac{dV}{dt} = [-g_L(V-V_{rest}) + I_{syn}(t) + I_0] / C. $$ According to Equation 2.13 of this paper,...
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70 views

Laplace transform of the zeroth-order Bessel function [duplicate]

Consider $J_0$ the zeroth order Bessel function. I'm trying to compute the Laplace transform $$\mathcal{L}[J_0](s) = \int_0^\infty J_0(t) e^{-st}dt,$$ but until now I couldn't find a good way to do ...