Questions tagged [laplace-transform]

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Upper bound for an inverse Laplace transform

Can anyone see how to get a tight upper bound for the function defined in terms of the inverse Laplace transform below? $$g_\text{sgd}(t)=\mathcal{L}^{-1}\left[\left(\frac{s}{2}+\frac{5 \sqrt{\frac{2}{...
Yaroslav Bulatov's user avatar
2 votes
1 answer
136 views

2D lattice sum with numerator

I've been struggling a bit with a double sum that arose as the trace of an operator: $$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$ where $n$ is an even natural number. Is there ...
R Grady's user avatar
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Which kind of convergence can we get from Laplace transform convergence?

This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that \begin{equation} \int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...
Fractional analysics's user avatar
1 vote
1 answer
124 views

Vague convergence VS Laplace transform convergence?

If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
Fractional analysics's user avatar
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19 views

Range of a Laplace-type transform

I'm interested in germs of functions $f(h)$ for $h\geq 0$ small. My question is, for what functions $f(h)$ can I write, for some $\delta>0$: $$f(h) = \frac{1}{h} \int_0^{\delta} e^{-s/h} a(s)\: ds +...
SnowRabbit's user avatar
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0 answers
84 views

Modeling decay of a linear system with a mixing term

I'm trying to analyze convergence of $x\in \mathbb{R}^{+d}$ which follows the following recurrence $$\mathbf{x}\leftarrow (\mathbf{1}-\mathbf{h})^2 \mathbf{x} + \mathbf{h}\langle \mathbf{x}, \mathbf{h}...
Yaroslav Bulatov's user avatar
1 vote
0 answers
35 views

Upper bounds on Inverse Laplace transform of a rational function

I need the inverse Laplace transform $\mathscr{L}^{-1}$ or a nice upper bound on $\mathscr{L}^{-1}$ for the following function: $$f(y)=\frac{\left(\sum_i \frac{u_i}{y-a_i}\right)^2}{1-\sum_i u_i \frac{...
Yaroslav Bulatov's user avatar
2 votes
1 answer
83 views

Injectivity of two sided Laplace transform

Let $\mu,\nu$ be finite Borel measures on $\mathbb R$. Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide: $$ \int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
Lauritz's user avatar
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1 answer
416 views

$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$

I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW ...
Wreior's user avatar
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What kind of functions can be represented as infinite linear combinations of exponential functions?

Let $f(x)$ be a real-valued function defined in $(0, \infty)$. I am curious what kind of $f(x)$ has the following representations: $$ f(x) = \sum_{j=0}^\infty a_j e^{-jx}, \quad \forall x \in (0, \...
fs l's user avatar
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1 answer
55 views

Discrete uniqueness sets for the two-sided Laplace transform?

Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by $$ Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx. $$ A ...
r_l's user avatar
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Monotonicity of a function defined by an integral

The question below is motivated by the related question Integral of a function changing sign and the associated answer: Can we study the monotonicity of the following function on $(0,1)$? $$\small f(x)...
Migalobe's user avatar
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1 answer
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Approximate inverse laplace transform in terms of the moments of a function

If $F(s)$ is the Laplace transform of $f(t)$ and \begin{equation} F(s)=\frac{1}{1-aG(s)} \end{equation} where $G(s)$ is the Laplace transform of a known probability density distribution $g(t)$ whose ...
Sam's user avatar
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Asymptotic expansion of the laplace transform for $s\to\infty$

Let $F(s)$ denote the Laplace transform of $f(t)$. Expanding for large $s$ one can obtain the following relation: $$F(s)\approx f(0)/s+f'(0)/s^2+2!f''(0)/s^3+3!f'''(0)/s^4+...\quad s\gg1$$ See e.g. a ...
Sam's user avatar
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3 votes
1 answer
277 views

Integral of a function changing sign

By some numerical tests, we can see that the following function is negative on $(0,1)$: $$\small f(x)=\int_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-...
Migalobe's user avatar
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2 votes
0 answers
99 views

Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform

Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
Mirar's user avatar
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329 views

Hardy–Littlewood Tauberian theorem for Laplace transform

The Hardy–Littlewood Tauberian theorem for Laplace transform in Chapter XIII in "An Introduction to Probability Theory and Its Applications" by Feller reads as follows Let $F : [0,\infty) \...
mnmn1993's user avatar
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6 votes
2 answers
314 views

On frequency decay of an integral transform of a function

Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that $$ \bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$ for all $\tau \...
Ali's user avatar
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11 views

What does $\limsup_{n\to\infty} |\mathcal{L}\mu(\theta+\mathbf{i}n)|<1$ mean?

Let $\xi$ be a point process, $\mu$ its intensity measure, i.e. $\mu(\cdot)=\mathbb{E}[\xi(\cdot)]$, and $\mathcal{L}$ the Laplace transform of $\mu$, i.e. $\mathcal{L}\mu(z)=\int_{0}^{\infty}e^{-zx}\...
toni_iva's user avatar
1 vote
0 answers
142 views

Discretizing a differential operator which is a function of the derivative operator

Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$. As an example, let $p(x)=x^{2}+2x$. Then ...
Mirar's user avatar
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5 votes
2 answers
161 views

Limit of the extremal process of i.i.d. Gaussians see from the tip

I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
MikeG's user avatar
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1 vote
1 answer
69 views

Initial and final Theorem for upper and lower limits?

Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$. Can we use upper limits or lower limits to ...
Fractional analysics's user avatar
6 votes
1 answer
152 views

Convergence speed of the tail of distribution using Tauberian remainder theorem

This question may be related to this one. Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem. Let $f$ be some one-sided probability ...
Seung Hyeon Yu's user avatar
-1 votes
1 answer
56 views

Finding the nth value of a dual series [closed]

I have a problem which i remember solving using Z transform in my uni. time, but i don't recall the EXACT way. I have 2 series of numbers: X[n] and Y[n] X[0] = Y[0] = 1 X[n+1] = aX[n] + bY[n] Y[n+1] = ...
Guy Barash's user avatar
0 votes
1 answer
93 views

Unique zero solution to a difference equation via Laplace transform

We want to prove that the unique solution to the following difference equation is the null one: $$ au(x)+b\mathbf{1}_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}_{(\frac{1% }{2},1)}(x)u(x-\frac{1}{...
Gustave's user avatar
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1 answer
153 views

Laplace transform injectivity for different values of $p$

Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty ).$ Assume that there exist $p_{0},p_{1}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ,$ $p_{0}\neq ...
Gustave's user avatar
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1 vote
0 answers
34 views

Solving an equation containing Laplace transform

Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where $\mathcal{L}$ is the ...
Goga's user avatar
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2 votes
1 answer
408 views

What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...
qifeng618's user avatar
  • 706
1 vote
1 answer
143 views

Inverse Laplace transform of $\frac{1}{s^a + 1}$ with $0 < a \leq 1$

Problem I am looking for the following inverse Laplace transform, $$ f(t) = \mathcal{L}^{-1}\left[\frac{1}{s^a + 1}\right] \;\;\;\;\; \text{with} \;\;\;\;\; 0 < a \leq 1. $$ What I understand ...
Keisuke FUJII's user avatar
4 votes
2 answers
408 views

Is it possible to use the Laplace Transform to calculate eigenvalues?

The relationship of Eigenvalues with Gradient Descent Let $A$ be a symmetric (and thus diagonalizable) matrix, with diagonalization $$A=VDV^T.$$ Let us define the quadratic function $$f(x) = x^T A x.$$...
Felix B.'s user avatar
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8 votes
0 answers
435 views

Modern treatment of Delange's Tauberian Theorem

Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper: H. Delange - Généralisation du théorème de ...
Daniel Loughran's user avatar
2 votes
0 answers
210 views

What is the role of the Laplace transform in the topological recursion formalism?

While reading papers on topological recursion, among them The Laplace transform, mirror symmetry, and the topological recursion of Eynard–Orantin by M. Mulase, they describe the mirror symmetry ...
rgvalenciaalbornoz's user avatar
3 votes
0 answers
105 views

Inverse Laplace transform through contour integration

How can I prove that in formal way, this function doesn't have inverse Laplace transform. $$ F(s)=\frac{\sin(s)}{\sqrt{s}} $$ Strictly it should be in Bromwich contour method. Could you please tell ...
meli0das's user avatar
4 votes
1 answer
2k views

Why is it possible to use the Inverse Laplace transform to get CDF?

I just saw the following on wikipedia about Laplace transformations: "In probability theory and applied probability, the Laplace transform is defined as an expected value. If $X$ is a random ...
Linus's user avatar
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1 answer
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Laplace transform and Laguerre Polynomials

What is the kernel $K(t)$ of the following Laplace transform equation: $$\int_{0}^{+\infty}e^{-(x+y)t} K(t) dt= \sum_{n=0}^{\infty}\varphi_{n}^{\alpha}(x)\varphi_{n}^{\alpha}(y),$$ where $\varphi_{n}^{...
Adam Hammam's user avatar
1 vote
1 answer
829 views

What is the inverse Laplace transform of $\frac{(1/s)_{n}}{s} $?

Introduction So far, I have found (p. 5) the following generating functions of the unsigned Stirling numbers of the first kind: \begin{equation} \tag{1} \label{1} \sum_{l=1}^{n} |S_{1}(n,l)|z^{l} = (z)...
Max Muller's user avatar
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2 votes
0 answers
138 views

Regularization of the area under hyperbola

So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
Anixx's user avatar
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1 vote
0 answers
57 views

Fractional power of the operator $\mathcal{L}_t[t f(t)](x)$ and equivalence of divergent integrals

I wonder whether an expression for fractional power of operator $\mathcal{L}_t[t f(t)](x)$ that involves Laplace transform can be derived? I am asking this because this operator preserves the area ...
Anixx's user avatar
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2 votes
2 answers
261 views

Laplace transform calculation

Please can someone help me? I have tried to find the Laplace transform of the form: $$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$ ...
Adam Hammam's user avatar
3 votes
0 answers
118 views

Solving a integro-differential equation

I am trying to solve an integro-differential equation: $$ {\frac{d}{dt}} f(t)=\int_0^t k(t-\tau)S(\tau)f(\tau) d\tau $$ with initial condition $f(0)=1$ If $k(t)=c\delta(t)$ with $c$ being constant, ...
J.G. Kang's user avatar
1 vote
2 answers
247 views

Can we meaningfully ascribe values to these divergent integrals?

My gut feeling is that $\int_0^\infty (1-\frac1{x^2})dx=0$ $\int_0^\infty (x-\frac2{x^3})dx=0$ $\int_0^\infty (x^2-\frac6{x^4})dx=0,$ etc, and in general, $\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$ ...
Anixx's user avatar
  • 8,838
0 votes
1 answer
115 views

Can I express this random variable in terms of known distributions?

By computing the Laplace transform of the total length of a random tree (the nested Kingman coalescent tree with coalescence rates $\gamma$ for the individuals and $\gamma'$ for the species), we would ...
Josue's user avatar
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2 votes
1 answer
436 views

Is inverse Laplace Transform of a power of $s$ a positive function?

It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace ...
Quiet_waters's user avatar
1 vote
1 answer
759 views

Inverse Laplace Transform using contour integration

So math stack exchange isn't really helping much with this. So initially, I'm proving the inverse laplace transform using contour integration. This is a good starting point for my research when I ...
Christoph's user avatar
1 vote
1 answer
87 views

Laplace transform of the product of two gammas

Suppose that $X$ and $Y$ are both gamma distributed with shapes $a,b$ and unit scales / unit rates. To fix ideas, X has Laplace transform given by: $$L_X(t) = \mathbb{E}(e^{-tX}) = (1 + t)^{-a}$$ How ...
lrnv's user avatar
  • 653
1 vote
0 answers
61 views

Laplace inversion with residue theorem doesn't satisfy IC of IVP

I have the following initial value problem $$ \frac{d\theta}{dt} = A(p-\theta) + B(\omega-\theta) $$ subject to the initial condition $$ \theta(0)=\theta_0 $$ and the constitutive set of equations $$ \...
Sharat V Chandrasekhar's user avatar
0 votes
0 answers
105 views

Laplace transform of sum of random variables in first hitting time problem

Let me refer to the example here. Suppose $X$ is a birth-death (BD) process (represents population size) that evolves by: $X \to X+1$ if a birth occurs with rate $\mu$, $X \to X-1$ if a death occurs ...
user36706's user avatar
0 votes
1 answer
332 views

Laplace transform inversion

I have a probability distribution that is defined through it's Laplace transform by : $$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$ Using R and the invLT package, i have a numerical ...
lrnv's user avatar
  • 653
1 vote
0 answers
116 views

About the computation of the inverse Laplace transform [closed]

I have several questions about the inverse Laplace transform: If $F(s)$ is a smooth real-valued function vanishing on a large subset of $\mathbb{R}$ (e.g. $F(s)$ is supported on a bounded interval), ...
Right's user avatar
  • 187
0 votes
1 answer
58 views

Laplace transforms of fractional equation

is there a finite expression of the Laplace transforms of the function \begin{align} L\left[ {\frac{{{x^n}}}{{{{(1 + x)}^m}}}} \right]\quad n \ge m \end{align}
hichem hb's user avatar
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