# Tagged Questions

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### Compute the following Laplace transform [closed]

I'm trying to Laplace transform the function $$|\theta(t)|\sin(l\theta(t)),$$ where $\theta(t)$ is any function of t. I want to express the result with $\tilde{\theta}(s)$, ...
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### Inverse Laplace transform of matrix exponential

I have the following Laplace-transformed, matrix-valued function: $$U(s) = e^{As + B},$$ where $A$ and $B$ are diagonalizable, noncommuting (but very close to commuting, if that's useful -- $B$ is ...
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### Solving a general, constant-coefficient, first-order, two-indep-variable system of PDEs

I have the following system of PDEs that I want to solve as "analytically" as possible: $$\left(\partial_t + A\partial_x + B\right)\mathbf{u}(t, x) = 0,$$ where $A$ and $B$ are constant, ...
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### How to find the Inverse Laplace Transform of the following?

I have a Laplace tranform in the form given below $\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$ Can some one help me to find ...
1answer
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### Inverse Laplace transform of a hypergeometric function

This is a repost from Math Stack-exchange where I did not manage to get an answer. http://math.stackexchange.com/questions/1491027/inverse-laplace-transform-of-a-hypergeometric-function I managed to ...
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### Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as $$\mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx$$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
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### Parseval's theorem

In operational calculus there is Parseval's theorem, which states that if $f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p)$ and both $F(p)$ and $\Phi(p)$ are analytical in $Re p \geq 0$, ...
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### Sum of two independent random variables

Let $\xi, \eta, \eta'$ be non-negative random variables such that: $\eta \stackrel{\mathcal{L}}{=} \eta'$, $\xi + \eta \stackrel{\mathcal{L}}{=} \xi + \eta'$, $\xi$ and $\eta'$ are independent. ...
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### a limit of the laplace transform and its derivative

If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
1answer
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### Integral representation of the resolvent of a semigroup

Let $T(t)$ be a $C_{0}$-semigroup with the generator $A$. Now, does the so called integral representation of the resolvent $$(\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt$$ hold for ...
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### Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...
1answer
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### Extracting moments from a special Z-transform

Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that \frac{X_n-n\mu}{\sqrt{n}\sigma}...
1answer
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### Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) \begin{...
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### How do I estimate/bound the error in an inverse Laplace transform?

Suppose I have a Laplace transform, $$F(s) = \int_0^{\infty}dx\ f(x)e^{-s x} \ .$$ I know that $$F(s) \approx e^{A/(4s)}$$ (for $s$ real) where $A$ is very large, and I want to estimate $f(x)$...
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### dominant pole in the laplace transform

hi, I have a function $X(t)$ whose Laplace transform $\hat{X}(s)$ has a unique pole of largest real part $x_0$, which is a real number. I am able to show that for each $t$, $X(t)$ is a convergent ...
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### Wiener-Hopf Integral/Lindley's Equation

Lindley's equation is well known within queueing theory and is as follows $F(y) = - \int_0^\infty F(x)dH(y-x)$ However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which ...
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### Laplacian operator and relation to the Laplace Transform

I'm trying to understand why the Laplacian operator is used in blob detection in image analysis. I must admit that in trying to figure out why the Laplacian is useful in this application, I've really ...
1answer
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### method of moments and Laplace transform from Shepp and Lloyd

Again from the Shepp and Lloyd paper "ordered cycle lengths in a random permutation", I found this puzzling equality. This one might require access to the paper itself since it's quite a mouthful: In ...
6answers
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### Fourier vs Laplace transforms

In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...
2answers
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### Can we extract information about how fast a function decay from its Laplace transform?

My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform. More concrete case, let \$f:\mathbb{R} ...
5answers
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### Applied mathematics Books (graduate level)

What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems? There is a lot of books on ...
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### Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ansätze do you know ...
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### Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...