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Inverse Laplace Transforms of Exponential Form

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
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1answer
82 views

inverse Laplace transform of the determinant

what is the inverse Laplace transform of the function $x\mapsto\det x$? (where $x\in\mathbb{R}^{n\times n}$ is a symmetric metric, if necessary at all.)
3
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1answer
189 views

Variations on the Mellin and Dirichlet transforms

There are a number of variations on the Laplace transform that turn up all over math. Some examples: $\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform $\sum_{-\infty}^{\infty} ...
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39 views

Functions whose Laplace transforms have prescribed behavior at minus infinity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non-negative function with entire Laplace transform $\hat{f}$ (in particular $\lim_{t\to \infty}e^{st}f(t)=0$ for all $s$), and $p_0$ a positive integer. ...
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0answers
32 views

How to find Laplace Transform of fractional-order differential systems

Let us consider the fractional-order complex-valued dynamic system as $D^\alpha z(t)=-Az(t)+Bz(t-\tau),\ z\in \mathbb{C}$. How to find the characteristic equation and the Laplace transformation?
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2answers
149 views

compute the limit of a rational function

Suppose I have a rational function defined by ($s$ complex) $$ f(s)=w^T s(sI-Q)^{-1} v $$ for nonzero column vectors $w,v$ and a (large) square matrix $Q$. Further assume that $Q$ is singular and that ...
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0answers
126 views

Existence of zero-free strip of a Laplace transform (edited ..)

Problem Let $\beta$ be a probability measure on $\mathbb{R}$, and define $$ K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is ...
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0answers
99 views

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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0answers
306 views

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 $, ...
2
votes
2answers
525 views

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. ...
1
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2answers
102 views

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 ...
3
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1answer
133 views

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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0answers
344 views

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 ...
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1answer
108 views

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 ...
0
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1answer
288 views

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) ...
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2answers
211 views

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 ...
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1answer
205 views

An Integral Functional Equation

Let $f$ be a non-negative function supported and integrable on the positive real axis, such that $$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$ where $c[p]$ a number (functional) dependent on function ...
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1answer
240 views

Characterization of the Laplace Transform

One of the main properties of the Laplace transform is given by the convolution theorem. $$\mathcal{L}(f*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$ Question: Is there a full characterization of the ...
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1answer
508 views

Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
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4answers
3k views

Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} ...
4
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1answer
175 views

Relations involving Stirling numbers of second kind

While inverting a Laplace transform using Post's inversion formula I found the following expression: $$ \sum_{k=1}^n S^n_k \ x^k(\alpha)_k $$ where $S^n_k$ is a Stirling number of second kind and ...
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0answers
169 views

Inferring asymptotic behaviour from the dominant pole of the Laplace transform

Hi, I am reposting the following question with the hope that a more detailed description will lead to a more descriptive response: dominant pole in the laplace transform I have a vector function ...
3
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1answer
294 views

Using a quadratic kernel instead of a linear kernel in the Laplace transform

Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: ...
1
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1answer
283 views

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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0answers
232 views

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 ...
4
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2answers
200 views

On a certain generalization of the Laplace transform

Let $\alpha$ be a positive constant, $\mu$ be a Borel nonnegative measure in $\mathbb{R}^n_+$. We can define a transform $$ \tilde{L}\[\mu\](p) = \int\limits_{\mathbb{R}^n_+} e^{-(p_1 x_1 + \ldots ...
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0answers
1k views

An inverse Laplace transform involving Error function

Dear MOs, I need to calculate the inverse Laplace transform of the following function $$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$ I have checked, among many ...
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2answers
693 views

Understanding the inverse Laplace transform of a function with essential singularities

I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in ...
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2answers
2k views

How does the Laplace Transform work for circuit analysis? [closed]

I would like to understand how signals transformed from the time domain to the frequency domain for algebraic manipulation, can be transformed back to give solutions in the time domain. Knowing how to ...
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0answers
235 views

On the generalisation of the Laplace transform

I consider a measure transform $A$ given by $$ A\mu(x) = \int\limits_{\mathbb{R}^n_{+}} e^{-g(x,y)} \mu(dy) $$ where $g(x,y)$ is some positive smooth function, $\mu$ is a Borel measure. Is it a ...
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6answers
5k views

What does Mellin inversion “really mean”?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
2
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1answer
160 views

Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution

I am trying to calculate the Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution. I am ...
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0answers
328 views

Laplace transform of a stopping time for stochastic volatility models

Let $V_t$ be a solution of the SDE $$dV_t=V_t(rdt+\sigma_t dW_t) $$ where $\sigma_t$ satisfies some other SDE $$d\sigma_t=\alpha(t,\sigma_t)dt+\beta(t,\sigma_t)dW^{\\ \prime}_t $$ and $W_t$ and ...
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3answers
1k views

When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as: $f(x) ...
3
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0answers
300 views

inverse Laplace transform of $\delta_1(\cdot)$

Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...
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2answers
1k views

Getting a differential equation for a function from a functional equation of its Mellin transform

If $f$ is a locally integrable function then its Mellin transform $\mathcal{M}[f]$ is defined by $$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$ This integral usually converges in a ...
3
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2answers
3k views

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 ...
2
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1answer
550 views

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 ...
12
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6answers
46k views

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 ...
2
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2answers
658 views

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} ...
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5answers
4k views

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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2answers
535 views

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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6answers
6k views

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 ...
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9answers
9k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...