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

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inverse sine transform, laplace transform and convolution theorem

when we apply Laplace and Sine Fourier transform to semi-infinite beam equation with boundary conditions $y(0,t)=V(t),~~~y_{xx}(0,t)=0$, we have, $\bar{Y_{s}}(\zeta,s)={\sqrt{2 \over ...
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40 views

Inverse Laplace Transform with cosh -fick's law diffusion

I'm trying to demonstrate highly quoted result of the Fick's second law of diffusion for a composite sphere of radius R2 (whose inner core, 0 ≤ r < R1, and outer shell, R1 ≤ r < R2, have ...
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25 views

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

I have been desperately trying to find the inverse laplace transform using the complex inversion formula for this question. $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$ I have found it extremely ...
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1answer
29 views

Find inverse Laplace transform of $H(s)=\frac8{s^4+4}$

How can we find the inverse Laplace transform of the function $$H(s)=\frac8{s^4+4}?$$
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13 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
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35 views

How to find inverse laplace transform

$$ F(s) = \dfrac{6s+9}{s^2-10s+29} $$ How do you solve the inverse Laplace transform of this above equation?
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1answer
30 views

How to solve Laplace initial value problem

$$ y''+36y = f(t) $$ $$ f(t) = \begin{cases} 1, & \text{0 ≤ t < 8} \\ 0, & \text{8 ≤ t < ∞} \end{cases} $$ $$ y(0) = 0 $$ $$ y'(0) = 1 $$
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45 views

inverse laplace transform by using complex integral

given function $$f(s)=\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$$ and $$\int_{0}^{\infty}{\frac{e^{-xt}}{\sqrt{x}(x+1)}dx=\pi e^t {erfc}(\sqrt{t})}$$ my steps: ...
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1answer
11 views

tail limit of Laplace transform of a bounded random variable

Suppose that $X$ is a variable such that $0<X<m$. I would like to know some information on the behavior of the function $$\phi(p)=\frac{1}p \log E e^{pX} $$ when $p\to\infty$. Here are some ...
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1answer
20 views

Inverse Laplace transformation of (s^2-4s-2)/((s^2+2)^2)

I approached this problem as follow: $1.$ rewrote $(s^2-4s-2)$ into $(s-2)^2-6$ $2.$ Now break the function into 2 parts: $\frac{(s-2)^2}{(s^2+2)^2} + \frac{6}{(s^2+2)^2}$ the Laplace inverse ...
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How injective is the Laplace transform?

Denote the Laplace transform by $\mathcal{L}$, and assume $\mathcal{L}[f]$ and $\mathcal{L}[g]$ exist for some functions $f$ and $g$. Then we know that $\mathcal{L}[f*g]=\mathcal{L}[f]\mathcal{L}[g]$. ...
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All possible inverse laplace transform of $ (s+1)^2/(s^2+2s+1)$

Well this was the question given in our previous week's test: Find all possible Inverse Laplace Tranforms of $(s+1)^2/(s^2+2s+1)$ . I just expanded the numerator and so the function reduced to 1.And ...
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2answers
22 views

Rewriting solution in terms of hyperbolic trigs

I have to find the inverse laplace transform of: $\mathcal{L}^{-1}(\frac{s}{-8+2s+s^2})$ I found it was $\frac{2}{3}e^{-4t}+\frac{1}{3}e^{2t}$ But the question I'm asked is, determine $A,B,C,D$ ...
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1answer
41 views

Can anyone tell me the name of this laplace transform?

I need to apply this rule to solve a Laplace transform: $\mathcal{L(\frac{f(t)}{t})}=\int_s^\infty F(u) du$ I've been given a table on laplace transform "rules" but I don't know how to use this one. ...
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16 views

final value theorem in the presence of white noise

I apply the final value theorem to get the steady-state error with the presence of white noise and I just keep getting zero. To me, it seems wrong to have zero steady-state error when there is noise ...
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1answer
34 views

Inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$

What is the inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$? Using the residues, I can calculate the residues at $s_n=2n\pi i$, but I have problem in calculating residue at $s=0$. ...
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1answer
35 views

Inverse laplace transform in a physics problem.

This came up during a physics problem, where we need to find the inverse laplace transform of $$X(s) = \left( 1+ \frac{k}{ms^{3/2}}\right)^{-1} \left( \frac{c_1}{s^2} + \frac{c_2}{s} \right)$$ to ...
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1answer
38 views

Calculating improper integral

Does anyone know how to solve the following integral: $$I =\int_{0}^\infty \cos(t \mathrm{log}( x))\,\mathrm{e}^{-ax}\, \mathrm{d}x,$$ where $t$ and $a$ are real. Please show some intermediate ...
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Find the impulse response to a RLC filter

I have a serial RLC-filter for which I should first determine the transfer function and then the impulse response. I figured out that the transfer function is: $H(s)=V(s)/U(s)$ And my circuit has the ...
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Inverse Laplace transform of a particular function

I am interested in evaluating the following Bromwich integral $$\mathcal{I}(t)=\frac{1}{2\pi i}\int^{\gamma+i\infty}_{\gamma-i\infty}\frac{e^{zt}}{1+z^{\beta}}\,dz$$ where $\beta>1$ and ...
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1answer
38 views

Integration and Laplace-Stieltjes of a multiplied Weibull and Exponential distribution Function

I have a trouble for integrating a multiplied weibull and exponential distribution. The expression is as follows: $$ Y(t) = \int_0^t e^{-\lambda T}e^{-(T/\mu)^z}dT\,. $$ Then, I need to take ...
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25 views

Laplace Transform and Convolution of Three Functions

So I'm trying to solve this differential equation: $y'' - 4y' + 3y = f(t)\:\:\:$ with initial conditions $y(0) = y'(0) = 0$ using laplace transform. After taking the Laplace transform and doing ...
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21 views

Laplace transform of $ \frac{1}{C}\int_0^1i(t)dt$

I have the integral: $$ \frac{1}{C}\int_0^1i(t)dt$$ which I should transform with Laplace. There is a rule saying that $$ \int_0^ti(t)dt$$ has the transform $$ s^{-1}F(s) $$ can I use this to ...
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1answer
36 views

Show $\mathcal{L}\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty}F(u)du$ [duplicate]

Show for $\mathcal{L}$, the Laplace transform, that $$\mathcal{L}\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty}F(u)du.$$ I know that $\mathcal{L}\left\{ t^n f(t) \right \} = (-1)^n ...
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1answer
33 views

How to solve this Laplace transform? $f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$

Find the laplace transform of $$f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$$ The answer is $$\frac{1}{2(s+2)}+ \frac{1}{2} \frac{s+2}{s^2 + 4s + 40} - \frac{6}{(s-3)^3}.$$ This took me about an hour to ...
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Using calculus to find inverse functions

High schooler here. Last summer I taught myself a little bit of calculus, and I have been doodling about it. So I began writing some problems for myself, and one of them was this: Find the inverse ...
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18 views

Laplace transform the expression $\int_0^t(t-u)y(u)du$

I laplace transformed the expression $\int_0^t(t-u)y(u)du$ in Wolfram and it seems like the answer is just $\frac{Y(s)}{s^2}$. If I change the expression to this: $$ \int_0^t(t-u)y(u)du = t\int_0^t ...
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Evaluate the $I=\frac{1}{\pi}\int_0^{\infty}\frac{e^{-xt}\sin (a\sqrt{x})}{x}\,\mathrm dx$

I want to evaluate $$I=\frac{1}{\pi}\int_0^{\infty}\frac{e^{-xt}\sin (a\sqrt{x})}{x}\,\mathrm dx$$ It seems that the solution should be in the form of the error function and also it involves contour ...
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2answers
33 views

What will happen after Laplace Transform?

Consider the Laplace transform $\int_{0}^{\infty} e^{-px}f(x)\,dx$ Assume $f(x)=1$ , then the Laplace transform is $\frac {1}{p}$. Assume $f(x)=x$ , then the Laplace transform is $\frac {1}{p^2}$. ...
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27 views

Inverse of Mellin transform

I would like to invert the following Mellin transform $M(s)$ of a function $f(x)$ defined on $[0,a]$ with $a>0$ (or get the $x\rightarrow 0$ asymptotics): $$ M(s) = \frac{2a^s}{s-2(1-a^s)} $$ We ...
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63 views

asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...
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The existence of the laplace transform

I don't understand why the laplace transform of some function, say f(t), has to be "piecewise continuous" and not "continuous". Is "piecewise continuous" sort of like the minimum requirement? This ...
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inverse Laplace transform by integral

I've seen this formula for the inverse Laplace transform in several books: $$f(t)=\mathcal{L}^{-1}\{F\}(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\alpha -iT}^{\alpha +iT}e^{st}F(s)ds$$ where $f$ is ...
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21 views

Find the Laplace transform of integral(from 0 to x) sin(2t) dt

Find the Laplace transform of $\int_0^x\,\sin\,(2t)\,dt$ So basically, $$\int_0^x\,\sin\,(2t)\,dt = -\frac{1}{2}(\cos\,(2x) - 1)$$ So $$\mathcal{L}\{\cos\,(2x)\} = \dfrac{s}{s^2 + 4}$$ ...
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An identity derived from the Laplace transform

It seems that $$\int_0^t \int_0^l f(\tau) ~d \tau ~d l = \int_0^t z f(t-z) dz $$ since the Laplace transform of both sides is $F(s)/s^2$, where $F(s)$ is the Laplace transform of $f(t)$: the left-hand ...
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23 views

Fourier series: term-by-term Laplace transform.

Quick question: If a Fourier series is uniformly convergent should the term-by-term Laplace transform of the series equal the result of the periodic function theorem for the Laplace transform?
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1answer
30 views

Laplace transform of the autocorrelation of a wss random process

Consider a wide-sense-stationary random process $x(t)$. The autocorrelation function is $r(t-\tau):=E[x(t)x(\tau)]$. Let $S(s)$ be the Laplace-transform of $r(t)$. Can I compute $S(s)$ as ...
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29 views

Inverse Laplace Transform of exponential

Is it possible to compute the inverse Laplace transform of: $$\frac{1}{1-e^{-sa}}$$ where $a>0$ ?
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Are FT and LT both isomorphic?

As the following diagram:(from a textbook) Note: 1. L2: L2 space, H2: H2 space 2. The upper one is in t-domain; the lower one, f-domain 3. : the Laplas transform operator : the fourier tansform ...
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Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why?

Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why? The Bromwich integral is not covered in my course so I can't use it. I'm hoping and guessing that the answer is simple! ...
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1answer
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Any closed formula for $\mathcal{L}\big(u_c(t)\cdot f(t)\big) $?

As in the title, is there any closed form formula for such Laplace transform, with denoting $\mathcal{L} \ f(t)=F(s)$?
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Find $\mathcal{L}^{-1}\frac{s}{s^2-6s+9}$

It is easy to see that $\frac{s}{s^2-6s+9}=\frac{s}{(s-3)^2}$ and now I want to use use the convolution integral for $s\cdot \frac{1}{(s-3)^2}$. So I get this integral: $$\int_0^t \delta '(\tau)\cdot ...
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23 views

The Laplace transform of $\mathcal{L}(te^t \cos t)$

How do I find it? I know that $\mathcal{L}(e^t \cos t) =\frac{s-1}{(s-1)^2+1^2}$ but what is it when multiplied by $t$, as written in the title?
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1answer
41 views

Laplace Transform's phase delay

I have read this example about time shift of Laplace Transform somewhere. It used a unit step function that has been shifted along $x$ axis for $a$ unit. So, to find the Laplace Transform of it, ...
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21 views

Laplace transform of $\frac{\sin at}{t}$

Laplace transform of $\displaystyle \frac{\sin at}{t}$ My Attempt: Rule used: $\displaystyle L[\frac{1}{t}f(t)]=\int_{s}^{\infty}\bar f(s)ds$ So, $\displaystyle L[\frac{1}{t}\sin ...
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1answer
18 views

Laplace transform, Inverse Laplace transform

Let $(\mathcal{L}f)(s)$ be the Laplace transform of a piecewise continuous function $f(t)$ defined for $t\geq 0$. If $(\mathcal{L}f)(s)\geq 0$ for all $s\in\mathbb{R^+}$ does this imply that $f(t)\geq ...
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26 views

Laplace transform on a finite interval $f(t)= \int_0^1 e^{-xt} f(x) \, dx$

What is the name of this transform? It's basically the Laplace transform where we integrate over a finite interval. $$ F(t)= \int_0^1 e^{-xt} f(x) \, dx$$ Is it still just the Laplace transform? ...
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32 views

inverse laplace transform of $s/(s^2+6s+13)$

Hi can anyone help with this inverse Laplace transform $$s/(s^2+6s+13) $$ I tried to do partial fraction $s+3/(s+3)^2+4 - 2/(s+3)^2+4$, but then I don't know what to do next...
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1answer
14 views

hybrid function into one-line form

I came across a non-homogeneous ODE with the non-homogeneous term $g(t)$ defined by a few functions like this one below: $$g(t)=\left\{\begin{matrix} f_1(t), & 0\leq t<a\\ f_2(t), & a\leq ...