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The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.
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Convolution Equation Solving
Solving the equation $$y'(t)=\sin(5t)-25 \int\limits_{0}^{t} y(u)\,\mathrm du$$ with $y(0)=0$, we obtain the convolution $$y(t)=\sin(5t)\cdot g(t)$$ For some function $g(t)$. What is the value of $g\... integration ordinary-differential-equations laplace-transform convolution- 91
Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$
Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$. The answer from the text book is $f(t)=\cosh^2(t)+\cos^2(t)$. But my result is $2\cos^2(t)\cdot \cosh^2(t)$. Which is ... complex-analysis inverse laplace-transform residue-calculus inverse-laplace- 1
Inverse Laplace transform of $\dfrac{1}{s(e^s+1)}$
The original problem is to solve $$\mathcal{L}^{-1}\left\lbrace\frac{e^s}{s(e^s+1)}\right\rbrace.$$ Doing partial fractions $$\frac{e^s}{s(e^s+1)}=\frac{1}{s}-\frac{1}{s(e^s+1)}$$ the problem reduces ... ordinary-differential-equations laplace-transform inverse-laplace- 119
With what classes of functions the equality $\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx$ leads to paradoxes?
The following operators keep the area under the convergent integrals unchanged: $$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,... laplace-transform integral-transforms divergent-integrals- 7,742
get first ODE general solution using integrating factor and laplace transform
$y' + ay = h(t) , y(0)= b $ is the question. I get $e^{at}$ as integrating factor, but I don't know where it uses while doing laplace transform. ordinary-differential-equations laplace-transform- 1
Visual interpretation of the Laplace-transform
I was wondering if there is a visual interpretation for the Laplace-transform. For example, you can visualize integrals by sketching the area under the graph. That way it has a visual meaning. I’m ... complex-analysis laplace-transform visualizationIn what cases the transform $\mathcal{L}_t[t f(t)](x)$ preserves the ordering?
Let's consider the set of integrable functions $f:[0,\infty)\to(-\infty,\infty)$ with countable number of singularities. Let's define order in such a way that $f>g$ if and only if $\int_0^\infty (f(... laplace-transform integral-transforms divergent-integrals- 7,742
Laplace transform of $te^{-2t}\sin(2t)u(t-3)$
Laplace transform of $te^{-2t}\sin(2t)u(t-3)$ I do know the following properties of Laplace Transform: A) $t f(t) = \frac {dF(S)}{ds}$ B) $e^{at} f(t) = F(S+a)$ But from what I see a part of the ... laplace-transform- 45
What is inverse Laplace transform of Dirac delta?
I know that the Bromwich integral to get inverse Laplace transform, which is given by $$ \mathcal{L}^{-1}(F(s))=\frac1{2\pi i}\lim_{M\to+\infty}\int_{\sigma-iM}^{\sigma+iM}F(s)e^{st}ds $$ , $\sigma$ ... complex-analysis laplace-transform inverse-laplace- 5,089
Solving $\int_0^x\int_0^t f(u)\,du\,dt$ using Laplace transform
So I've been trying to solve this problem with Laplace transform. But the problem is I don't know the function and therefore couldn't even get close to the answer! $$\int_0^x\int_0^t f(u)\,du\,dt=\... laplace-transform multiple-integral- 1
Trouble with ODE by Laplace transform with boundaries
From section 5.2 of Zill's book Differential equation with boundaries problems, I have to resolve the next equation with Laplace transform \begin{equation} \frac{d^{2}}{dx^{2}}\left(EI\frac{d^{2}y}{... ordinary-differential-equations laplace-transform- 274
why is the Laplace transform of local integrable function with support on $[0,\infty)$ analytic?
There is a proposition about Laplace transform, but I don't know how to prove it. Let $f \in L^1_{loc}(\mathbb{R})$, $\operatorname{supp}(f) \subset[0, \infty)$, such that $a$ is the abscissa of ... complex-analysis uniform-convergence laplace-transform transformation analytic-functions- 431
What's wrong with this Laplace transform?
The following operators keep the area under the convergent integrals unchanged: $$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,... laplace-transform integral-transforms inverse-laplace divergent-integrals- 7,742
absolut convergence of the Laplace transform of a intensity measure
Let $\xi$ be a point process and $\mu$ its intensity measure, i.e. $\mu(\cdot)=\mathbb{E}[\xi(\cdot)]$. The Laplace transform of $\mu$: $\mathcal{L}\mu(z)=\int_{0}^{\infty}e^{-zx}\mu(dx)$ converges ... measure-theory laplace-transform absolute-convergence point-processes- 11
Modified Bessel function and laplace transform?
WILLING TO PAY Hi, I'm doing revision for Laplace transforms and I have absolutely no idea how to complete this question properly. THOUGHTS So I'm just following what they have told me to do and ... integration laplace-transform bessel-functions laplace-method- 103
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