Questions tagged [laplace-transform]

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Inverse Laplace if products of hyperbolic function to other function [closed]

I want to calculate $\sinh(as) F(s)$. We have $$L^{-1}\left [ \left ( \frac{e^{as}-e^{-as}}{2} \right )F(s) ​\right]=\frac{1}{2}L^{-1}\left ( e^{as} F(s)\right )-\frac{1}{2}L^{-1}\left ( e^{-as}F(s) \...
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129 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 ...
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5 votes
2 answers
144 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 ...
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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 ...
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6 votes
1 answer
103 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 ...
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-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] = ...
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71 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}{...
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1 answer
121 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 ...
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Solving a difference functional equation by using Laplace transform

Consider the operator $T:L^{2}(0,r+1)\longrightarrow L^{2}(0,r+1)$: \begin{equation*} Tu(x)=:u(x)+a\mathbf{1}_{(0,1)}(x)u(x+r)+b\mathbf{1}_{(1,1+r)}(x)u(x-1),% \text{ }x\in (0,r+1), \end{equation*} ...
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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 ...
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1 answer
276 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 ...
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gamma mixture representation of generalized Mittag-Leffler distribution

I am looking for a gamma mixture of the generalized Mittag-Leffler distribution. Question Let $f(x; \alpha, \nu)$ be the generalized Mittag-Leffler distribution. Its Laplace transform is $$ \mathcal{L}...
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1 answer
110 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 ...
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221 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 $$f(x) = x^T A x.$$...
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8 votes
0 answers
386 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 ...
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2 votes
0 answers
181 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 ...
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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 ...
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2 votes
1 answer
611 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 ...
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151 views

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}^{...
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1 vote
1 answer
678 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)...
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2 votes
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111 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 ...
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52 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 ...
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2 votes
2 answers
244 views

Laplace transform calculation

Please can someone help me? I have tried to find the Laplace transform of the form: $$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$ ...
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3 votes
0 answers
99 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, ...
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1 vote
2 answers
213 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, $\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$ ...
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1 answer
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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 ...
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2 votes
1 answer
261 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 ...
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1 vote
1 answer
526 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 ...
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1 vote
1 answer
53 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 ...
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59 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 $$ \...
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0 answers
66 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 ...
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0 votes
1 answer
263 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 ...
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1 vote
0 answers
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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), ...
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1 answer
56 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}
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3 votes
0 answers
165 views

Smoothness and decay correspondence for Laplace transform

For the Fourier transform, there are various theorems formalizing a correspondence between the smoothness of a function and the rate of decay of its Fourier transform. For example, if a function is $n$...
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6 votes
2 answers
178 views

Growth of (integral of) Laplace transform of a function of compact support as $Re \to -\infty$

Let $f:[0,\infty)\to \mathbb{R}$ be supported on $[0,1]$, with $\int_0^1 f(x) dx = 1$. Let $\mathcal{L} f$ be its Laplace transform. How slowly may $$\int_{-\infty}^\infty |\mathcal{L} f(\sigma+i t)| ...
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7 votes
1 answer
844 views

Where does the Laplace transform come from?

The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform. Q. What can we say concerning the Laplace transform?
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Is this integral transform related to the Laplace transform?

The Laplace transform of a function $f(t)$, defined for all real numbers $t \geq 0$, is the function $F(s)$, defined by $${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}.$$ Let $\varphi: {\...
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2 votes
1 answer
173 views

Integral transformation, Laplace-like

Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)? $$ \int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt $$ It resembles somewhat the Laplace transformation. ...
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2 votes
1 answer
750 views

Bromwich integral transformed to an integral on the real axis

I am new in complex integration and inverse Laplace transforms. I already asked this question on math.se but got no answer. The author of a textbook claims that the inverse Laplace transform has ...
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2 votes
1 answer
190 views

Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization)

I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also ...
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1 vote
0 answers
86 views

Is harmonic mean of linear functions a Bernstein function?

According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function: $f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$ is a Bernstein function, ...
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1 vote
0 answers
85 views

Expressing 1-e^{-z} as a Fourier integral

According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-...
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3 votes
1 answer
150 views

Asymptotics for an exponential generating function from an ordinary

I'm interested in taking an ordinary generating function $$F(x)=\sum_{n\geq 1}m_nx^n$$ and converting it to an exponential generating function $$M(x)=\sum_{n\geq 1}m_n\frac{x^n}{n!}.$$ I would then ...
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0 votes
1 answer
220 views

Weak continuity under Laplace transform

Let the sequence $u_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $U_n(s)$ and $U(s)$? $U_n(s)$ converges point-wise to $U(s)...
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270 views

Laplace transform of a random variable: Inversion formula from an interval

Let $X$ be a non-negative random variable with a CDF $F$. Let $L_X(t)$ denote the Laplace transform of $F$, i.e., \begin{align} L_X(t)=E[ e^{-tX}], \quad t \ge 0 \end{align} It is known that $L_X(...
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2 votes
1 answer
170 views

Can Mellin transform be applied in this function? What's the result?

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
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3 votes
0 answers
119 views

Decay of Laplace (or Mellin) transform beyond region of convergence?

Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
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1 vote
0 answers
166 views

Diffusion equation solution using Laplace transform [closed]

Consider the operator $$ L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t} $$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...
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1 vote
1 answer
85 views

Identity for stable Lévy subordinator

I want a proof or a reference for the identity $$ \int_0^\infty \frac{s^{n-1}}{\Gamma(n)} p_\beta(s,x)\,ds =\frac{x^{n\beta-1}}{\Gamma(\beta n)},\quad x>0, \,n\in\mathbb N, $$ where $x\mapsto p_\...
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