# Tagged Questions

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

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### A question on the Laplace Transform of $f(t)=t e^{at}\sin (bt)$ [on hold]

I would like to solve the Laplace transform of the following function: $$t \mapsto t e^{at}\sin (bt).$$ I know that $\mathscr{L}\left(\sin(bt)\right)=\dfrac{b}{s^2+b^2}$ and that you have to ...
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### Laplace transform in ODE

Use any method to find the laplace transform of coshbt Looking to get help with this example for my exam review
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### Laplace transform of bell-shaped functions

A real smooth function $\varphi$ is said bell shaped iff as the Gaussian : $\varphi''$ is positive on $(-\infty,a) \cup (b,+\infty)$ and negative on $(a,b)$. I'm interested in the bilateral Laplace ...
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### Analysing the modes of a signal with Laplace transform

If I have a linear dynamical system (assume continuous time for the time being) I can create the transfer function, let's say: $$\frac{1}{(s+a_1)(s+a_2)}$$ and the pole-zero map (this one is for e.g. ...
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### Evaluate $\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$ using residue theorem.

$$\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$$ I couldn't solve this problem using the residue theorem. Can anyone help me get the answer? I know the steps like taking the ...
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### Finding the inverse laplace transform using complex analysis.

I've been able to prove simple laplace transforms like $\dfrac {1}{(s+a)}$ quite easily but what about $\dfrac {1}{(s+a)^3+b^2}$ this does not seem easy to do since you cannot easily compute the ...
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### Criteria for $L^1$ convergence looking at Laplace transforms

Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ ...
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### Laplace transform of $\int_{0}^\infty\frac{e^{-t}\sin^2t}{t}dt$

Laplace transform of $\int_{0}^\infty\frac{e^{-t}\sin^2t}{t}dt$. So far I've calculated that $\frac{e^{-t}\sin^2t}{t}$ transformed equals $\frac{1}{8}(\ln((s+1)^2+4)-2\ln(s+1))$. My question is what ...
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### Rewriting $8H(t-\pi)(sint)$ without use of the heaviside function

I was given a differential equation to solve using Laplace transformation. and I got a term that had : $-8H(t-\pi)(sint)$ The question asks to rewrite the solution without the use of the heaviside ...
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### Simple way to prove that $e^{-x^2}$ doen't admit Laplace inverse trasform

The question is already contained in the title. Is there any criterion that one can use to show this or is it necessary to apply Mellin's inverse formula and verify that the integral doesn't converge?
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### Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
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### Laplace transform of the square root of a generic function

Let $f(t)$ be a function (for example of time $t$). Is there a general expression of the laplace transform of $\sqrt{f(t)}$ ? Same question for the inverse Laplace transform : Let $f(s)$ be the ...
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### Another Laplace transform of a function with square roots.

This question is very much related to this (one). Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{4+3s+\sqrt{s(4+s)}}.$$ My question is what is the inverse Laplace transform ...
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### Laplace transform of $\cos(at)$

I need to find the Laplace transform of $\cos(at)$ I know that $L\{\cos(at)\}= \int_{0}^{\infty} e^{-st} \cos (at) dt$ but I am having trouble finding the integral Thank you
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### Complicated Laplace Transform

I have found the following Laplace Transform in a list $$\int\limits_0^{\infty}e^{-st}\frac{e^{-u^2/4t}}{\sqrt{\pi t}}dt = \frac{e^{-u\sqrt{s}}}{\sqrt{s}}.$$ I am wondering how to prove this? I ...