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

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Find the Laplace transform?

Find the Laplace transform of $f(t) = 1 + (1 - t)u_1(t) + (t-2)u_3(t)$. Obviously each term of the function must be of $f(t - c)u_c(t)$ or be clearly transformable. Thus we have for our first term ...
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How to use Complex Inversion Theorem to find the Inverse Laplace Transform?

How to use Complex Inversion Theorem to find the Inverse Laplace Transform for the given $F(t)=L^{-1} \{s^{-1/2} e^{-1/s}\}$ ? Hint: make the radius $\epsilon$ of the inner circle $t-1/2$ rather than ...
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54 views

Showing an inequality

I wish to show $$|{(Re^{i \theta})^{-\frac{1}{2}}}\exp(\frac{-1}{Re^{i \theta}})| < \frac{M}{R^k}$$ for some M, k > 0 I've managed to reduce it to $$|R^{-\frac{1}{2}}| |\exp(\frac{-1}{Re^{i ...
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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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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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78 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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359 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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141 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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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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46 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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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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39 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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116 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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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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96 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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82 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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88 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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301 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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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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167 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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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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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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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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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: $u(t)= ...
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$\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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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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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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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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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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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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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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163 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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51 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 ...