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

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How to find the Laplace-Stieltjes transform of a joint distribution?

Consider two r.v.s (not necessarily independent) $X$ and $Y$ distributed exponentially with rate $\lambda$ and $\mu$ and having LSTs $E(e^{-sX})=\frac{\lambda}{\lambda+s}$ and $E(e^{-\theta ...
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90 views

Why is there an exponential in Fourier's defining integral?

I am having a hard time relating integration with Fourier series. Basically, I just get lost where there is an exponential in the integration to convert into the frequency domain. If someone can ...
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85 views

Find the inverse Laplace transform $f(t)=L^{-1}\left\{F(s)\right\}$ of the function $F(s)=\dfrac{7s−22}{s^2−6s+13}. $

Find the inverse Laplace transform $f(t)=L^{-1}\left\{F(s)\right\}$ of the function $F(s)=\dfrac{7s−22}{s^2−6s+13}. $ $f(t)=L^{-1}\left\{\frac{7s-22}{s^2-6s+13}\right\}$. I was trying to break ...
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421 views

Finding Inverse Laplace Transform using Taylor Series

Find the inverse Laplace transform $F(t)=\mathcal{L}^{-1}(s^{-\frac{1}{2}}e^{-\frac{1}{s}})$ using each of the following techniques: Expand the exponential in a Taylor series about s=∞, and take ...
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195 views

Inverse Laplace Transform for $F(s) = (9s-24)/(s^2-6s+13)$

Find the inverse Laplace transform of $\displaystyle F(s) = \frac{9s-24}{s^2-6s+13}$. I have tried factoring out a $3$ from the top and putting it into the form of $\displaystyle\frac{b}{(s-a)^2+b^2}$ ...
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32 views

Use the relation of Laplace Transform and its derivative to figure out $L\left\{t\right\}$,$L\left\{t^2\right\}$,$L\left\{t^n\right\}$

If $F(s) = L\left\{f(t)\right\}$, then $F'(s) = -L\left\{tf(t)\right\}$ Use this relation to determine $(a)$ $L\left\{t\right\}$ $(b)$ $L\left\{t^2\right\}$ $(c)$ $L\left\{t^n\right\}$ for any ...
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48 views

What is the name of this function similar to convolution?

The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example: $g(t,u) ...
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40 views

Verify that the transform of $y(t) = t^2e^{at}$ is $Y(s) = \frac{2}{(s-a)^3}$

I made the distinction to amplify "=" 3 times for easier readability. I tried: $$F(s) === \int_0^\infty t^2e^{(a-s)t}dt === \frac{1}{a-s}e^{(a-s)t}t^2\Big|_0^\infty \ - \frac{2}{a-s}\int_0^\infty ...
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40 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
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1answer
119 views

Laplace transform (differential equation containing several functions)

I have a differential equation which looks like: $$ \dfrac{dT}{dt} = \dfrac{P}{\rho Ac_ph} + \dfrac{q (T_{in} - T)}{Ah} - \dfrac{U\pi D(T - T_{a})}{\rho Ac_p} $$ where $P$, $h$, $q$, $ T_{in}$ are ...
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50 views

Laplace's Initial value theorem: discontinuity in $0$

The Laplace's Initial value theorem: $$\lim_{t\to 0}f(t)=\lim_{s\to \infty}sF(s)$$ This is a demonstration: $$ \lim_{s\to \infty} sF(s)=\lim_{s\to \infty}[\int_{0^-}^{\infty} \frac{d}{dt}f(t) ...
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140 views

How to solve this Reaction-Diffusion problem by FEM?

I want to solve this by Finite Element Method numerically, since the exact solution is too hard. Separation of variables does not help me here. Epsilon is positive so cannot be Helmholtz equation. ...
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1answer
102 views

Inverse Laplace Transform Problem

I have this problem $\frac{1}{(s^2+1)^3}$. I have to find its Inverse Laplace Tranformation. I already try using partial fraction but it didn't work because I found it will back to the problem form. ...
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3answers
93 views

How does the author get from one step to another?

I have to apply convolution theorem to find the inverse Laplace transform of a given function. I know that convolution is applied when the given function is multiplication of two functions. The ...
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473 views

find the Laplace transform of cos(sqrt(t))/sqrt(t) [closed]

$$ L\{ \frac{cos(\sqrt{t})}{\sqrt{t}}\} $$ Laplace-transform
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138 views

Show that $\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} =\ln\frac{b}{a}$ [closed]

It should be using Laplace transform. I found similar problems already solved but I need this to be shown using Laplace transforms: $$\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} = \ln\frac{b}{a}$$
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1answer
210 views

Inverse Laplace Transform via residues

I have $\frac{1}{2 \pi i} \int_{\infty-iT}^{\infty+iT} \frac{e^{-s(1-t)}-e^{st}}{-s+e^{-s}-1} ds$ and I am trying to solve it using a contour. So I could have t>0, or t<0. I have a pole at 0. For ...
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1answer
24 views

Representing an equation in the laplace domain

The equation below represents how the conductance of a sensor changes with respect to a change in carbon dioxide level: $$\text{Conductance} = A + Bx - Bx e^{-Ct}$$ where $A,B,C$ are constants, ...
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122 views

Wave equation 1D inhomogeneous Laplace/Fourier Transforms vs Green's Function

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 ...
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2answers
98 views

Why the Laplace transform of the integral is 1/s?

I got this interpretation: If we have $y'(t)=u(t)$ , it's like have $y(t)=\int{u(t)dt} $ If we solve this simple equation, we obtain: $$sY(s)=U(s)$$ $$Y(s)=\frac{U(s)}{s}$$ So, we have the $U(s)$ ...
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61 views

Extracting equation from a graph and Laplace's delay property

Good day to everyone, I tried to solve this problem but I'm not sure about the solution I chose. I had this function in Input of my system: I thought that the input was like this: $u(t)= ...
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121 views

$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ is that correct?

Good day. This integral looks very simple, yet I don't know how to start. $$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx$$ I know that if the lower integration limit was $-\infty$ it would ...
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157 views

Laplace transformation of $ \frac{\sin^2{t}}{t^2} $?

I tried convolution theorem also tried this process.. $$Laplace(\frac{f(t)}{t}) = \int_s^\infty F(u) du $$ So, $$Laplace(\frac{\sin{t}}{t} * \frac{\sin{t}}{t}) = \int_s^\infty F(u) du * \int_s^\infty ...
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357 views

Inverse Laplace of $ \frac{1}{\sqrt{s} - 1} $?

please help with this. I found this in textbook. Not derived from any differential equation. Also found the answer $$ \frac{1}{\sqrt{\pi}\sqrt{t}} + e^t * erf(\sqrt{t}) $$ (but don't know how)
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47 views

laplace transform and infinitely differentiation

This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} ...
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2k views

Is impulse response always differentiation of unit step response of a system?

I was trying to solve a question in which the transfer function of a system was asked, its unit step response being given: c(t) = 1-10exp(-t) The method that the book followed was to first find out ...
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58 views

Computing the inverse Laplace transform of this?

What's the correct way to go about computing the Inverse Laplace transform of this? $$\frac{-2s + 1}{(s^2+2s+5)}$$ I Completed the square on the bottom but what do you do now? $$\frac{-2s + ...
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71 views

How do I go about performing the following Laplace transform?

I'm unable to compute the following Laplace transform. How do I deal with cases such as $$f(t) = \sin(t-3)\theta(t) \quad \text{or} \quad f(t) = \sin(t-3)\theta(t-3),$$ where $\theta(t)$ is the ...
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3answers
189 views

Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
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897 views

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

My question is this: 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)= ...
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186 views

Laplace From Fourier transform?

This video (no need to actually watch it) makes a great point. If we interpret the $f$ in $f(x)$ as a function of time, then the fourier transform of $f$ takes the representation of this function in ...
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78 views

How do I perform Inverse Laplace on this function?

$$ F(S) = \frac{-S+11}{S^2-2S-3} $$ Howo do I find $f(t)$? What is a good strategy for attacking these types of problems? Thanks a bunch in advance for your help!
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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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72 views

Laplace transformation problem

There is a timely unchanged continuous function : $$H(s)=\frac{s-1}{s+1}$$ At the entry of the system exists a $x(t)$ which Laplace's transformation is: $$X(s)=\frac{(5s^2 - 15s + ...
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240 views

Use Laplace transform to solve the following initial–value problems.

Use Laplace transform to solve the following initial–value problem. $y′′′′ + 2y′′ + y = 0, y(0) = 1, y′(0) = −1, y′′(0) = 0, y′′′(0) = 2$ Answer $s^4 L(s) - s^3y(0) -s^2 y'(0) - s y''(0) - y'''(0) ...
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108 views

Solving initial value problem using Laplace Transform

Use Laplace transform to solve the following initial–value problems. a). $y'' + y = e^{−t}\cos 2t, \\ y(0) = 2, y′(0) = 1$ After using the concept of partial fraction and using Elementary Laplace ...
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534 views

Power series for the Bessel function using Laplace transforms?

The Bessel's function of the first kind of order zero, $J_0$ is the solution to $$ty''+y'+ty=0$$ which satisfies $J_0(0)=1$ The Laplace transform of this equation gives ...
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42 views

Laplace transform restriction and differentiation

every one.I have just started learning Laplace transform.However, there are two main conceptual problems I can't convince myself. The first problem is about the restriction of this integral, I ...
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3answers
52 views

help with laplace transform

Can you please help me with this Laplace transform? I used wolfram alpha to get the answer but I need some hints about the procedure to get to that answer. $$ \mathcal{L}\left(\frac ...
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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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344 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
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41 views

inverse laplace transform - with symbolic variables

Transform: $$ F(s) = \frac{2s^2 + (a-6b)s + a^2 - 4ab}{(s^2-a^2)(s-2b)} $$ My steps: $$ F(s) = \frac{2s^2 + (a-6b)s + a^2 - 4ab}{(s+a)(s-a)(s-2b)} $$ $$ = \frac{A}{s+a} + \frac{B}{s-a} + ...
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188 views

Laplace Transforms and third-order derivatives

The question is to calculate the Laplace transform of $(1 + t.e^{-t})^3$. I know that this can be done using a property where the problem is of the form of $t.f(t)$. However, I seem to be messing up ...
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34 views

Laplacian for Radon inversion theorem

Can someone check my proof in regards to the inversion of the Radon transform in $\mathbb{R}^2$ and $\mathbb{R}^n$. define $(-\Delta)^a f(x) = \int_{\mathbb{R}^d} (2\pi|\xi|)^{2a} \hat{f}(\xi)e^{2\pi ...
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47 views

Stuck on laplace transform question

I have to solve the following initial value problem using the laplace transformation: $$y'' + 4y = 0$$$$y_0 = c_1, y'(0) = c_2$$ I have taken the laplace transform of both sides, then rearranged ...
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87 views

Partial fractions for inverse laplace transform

I have the following function for which I need to find the inverse laplace transform: $$\frac1{s(s^2+1)^2}$$ Am I correct in saying the partial fraction is: ...
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57 views

Evaluating Laplace Transform

I have a Laplace transform function of the following form and I'm trying to evaluate it. From my research I think I need to take the Inverse Laplace Transform and then integrate, but I'm having ...
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51 views

Evaluating integration with Laplace transform

I am taking a differential equation class and for Laplace transformations and I have to find $$\displaystyle \int_0^\infty \dfrac{\sin t}{t}dt.$$ How can I do that?
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110 views

Laplace transform and integration together

The question, given in the textbook, is somewhat different. However, I am rephrasing it as follows: $$ \frac{2}{\pi}\int_{0}^{\frac{\pi}{2}} \left[ \mathcal L \lbrace \cos(t\cos\theta) \rbrace ...
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88 views

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} ...