Questions tagged [laplace-transform]

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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
234 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
114 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
5 votes
1 answer
3k 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
265 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}^{...
Adam Hammam's user avatar
1 vote
1 answer
889 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
164 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
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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
276 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
177 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
273 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
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0 votes
1 answer
118 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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3 votes
1 answer
601 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
844 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
108 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
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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
138 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
419 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
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0 answers
140 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
64 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
  • 367
4 votes
0 answers
393 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 $n$...
Sridhar Ramesh's user avatar
6 votes
2 answers
216 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)| ...
H A Helfgott's user avatar
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7 votes
1 answer
1k 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?
ABB's user avatar
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1 answer
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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: {\...
Into's user avatar
  • 1
2 votes
1 answer
180 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. ...
borntomath's user avatar
2 votes
1 answer
1k 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 ...
Stéphane Laurent's user avatar
3 votes
1 answer
310 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 ...
Learning math's user avatar
1 vote
0 answers
92 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, ...
Timothy Chu's user avatar
1 vote
0 answers
96 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)-...
Timothy Chu's user avatar
3 votes
1 answer
213 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 ...
Colin Defant's user avatar
0 votes
1 answer
320 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)...
Saj_Eda's user avatar
  • 395
1 vote
0 answers
435 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(...
Boby's user avatar
  • 631
2 votes
1 answer
182 views

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

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
Dimitris Valianatos's user avatar
3 votes
0 answers
135 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 ...
H A Helfgott's user avatar
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1 vote
0 answers
217 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 $u(x,0)=g(x), \forall x\in \Bbb R$...
Jim Art's user avatar
  • 111
1 vote
1 answer
101 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_\...
Rgkpdx's user avatar
  • 213
1 vote
1 answer
230 views

Laplace transform of the tetration (integral or series)

How to get some insight in the following integral: \begin{equation} \mathcal{I}(s)=\int_0^\infty x^{-x}e^{sx}\text{d} x \end{equation} where $s$ is real (and the lower integration bound may be set ...
Alexandre's user avatar
  • 368
3 votes
0 answers
68 views

Limits of a simple damped system

Definition: Let $F_n(s) = \frac{1}{s^{n+1}(1+s)^n}$ be the Laplace transform of $f_n(t)$. Required Result: To show $\lim_{n\rightarrow\infty}f_n(n+n/e) < o(n)$. Ideas: Let $G_n(s)=\frac{1}{s^{n+...
Vivek Bagaria's user avatar
3 votes
1 answer
121 views

Propagation error for ODEs

I am looking for a generic estimate to the following problem coming from biology: I am solving the ODE $$y'(t)=Ay(t)+zf(t), y(0)=0.$$ where $f$ is an external force determined by us and $z$ a ...
T.Sell's user avatar
  • 31
2 votes
0 answers
67 views

Question about a set of Laplace-transforms

A couple of years I asked in MSE about a set of Laplace-transforms getting no answer so far but got curious again yesterday. I'm putting the question here and as much focused as possible, just as a ...
Gottfried Helms's user avatar
0 votes
1 answer
209 views

How to numerically invert a bilateral (two-sided) Laplace transform?

For one-sided Laplace transforms I can find many algorithms to invert them numerically (e.g. algorithms named after: Talbot, Stehfest, Euler, ...). However, I am interested in numerical inversion of ...
David's user avatar
  • 1
4 votes
1 answer
324 views

Extended convolution theorem for Laplace transform

Let $(f*g)(t):=\int_0^t f(s) g(t-s)ds.$ Then the Laplace transform $L$ satisfies $L(f*g)(t)=L(f)(t)L(g)(t).$ This is known as the convolution theorem. I would like to know whether something similar ...
Zinon's user avatar
  • 41
0 votes
1 answer
211 views

What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
user avatar
2 votes
1 answer
620 views

The Borel-Laplace transform of a transeries that contains logarithms

I am interested in Ecalle's generalization of the Borel-Laplace summation. I would like to see an explicit treatment of a summation of a transeries that include logarithmic terms. The only example I ...
tst's user avatar
  • 483
1 vote
1 answer
123 views

representation of the Wright function

The two-parameter Wright function http://dlmf.nist.gov/10.46 is defined as the infinite series $$ \phi (\alpha, \beta \, | z)=\sum\limits_{k=0}^\infty \frac{z^k}{\Gamma(k+1) \Gamma(\alpha k + \beta) ...
Dimiter P's user avatar
  • 203
3 votes
2 answers
486 views

Does the hitting time of +1/-1 of a Brownian motion posess a density?

The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$. I define $T=\inf \{t>0,|W|(t)= 1\}$. ...
Adrien Laurent's user avatar
1 vote
1 answer
1k views

Inverse Laplace transform to get CDF

I have the following problem. If I can get some help, I would greatly appreciate it. I am trying to replicate a particular research paper and came across a problem: Suppose $X$ is a birth death ...
turtle_in_mind's user avatar
16 votes
4 answers
1k views

Proof of complete monotonicity of a binomial function

By plotting the function and its derivatives, one can easily be convinced that the function $$f(x):=\log\binom{x}{p x}=\log\Gamma(x+1)-\log\Gamma(px+1)-\log\Gamma((1-p)x+1),$$ defined for $x>0$ and ...
MCH's user avatar
  • 1,304
1 vote
0 answers
92 views

How to solve differential equation for cylindrical diffusion?

How the differential equation for diffusion along a hollow cylinder, $$ \frac{\partial c}{\partial t} = D \Biggl(\frac{1}{r^2}\frac{\partial^2 c}{\partial \phi^2}\ + \frac{\partial^2 c} {\partial z^2}...
Mona's user avatar
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2 votes
0 answers
189 views

Under which conditions could a function analytic on a right half-plane be a unilateral Laplace transform of a function?

What are the necessary and/or sufficient conditions for a function holomorphic on a right half-plane to be a unilateral Laplace transform of ANY function, square integrable or not, for which the ...
Just A Young Artist's user avatar