# Questions tagged [laplace-transform]

The laplace-transform tag has no usage guidance.

123
questions

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votes

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24
views

### 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) \...

1
vote

0
answers

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 ...

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 ...

0
votes

1
answer

56
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 ...

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 ...

-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] = ...

0
votes

1
answer

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}{...

0
votes

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 ...

0
votes

0
answers

34
views

### 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*}
...

1
vote

0
answers

32
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 ...

1
vote

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 ...

0
votes

0
answers

22
views

### 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}...

1
vote

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
...

0
votes

1
answer

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
$$f(x) = x^T A x.$$...

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 ...

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 ...

3
votes

0
answers

84
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 ...

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 ...

0
votes

1
answer

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}^{...

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)...

2
votes

0
answers

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 ...

1
vote

0
answers

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 ...

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:
$$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$
...

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, ...

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,
$\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$
...

0
votes

1
answer

106
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 ...

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 ...

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 ...

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 ...

1
vote

0
answers

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
$$
\...

0
votes

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 ...

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 ...

1
vote

0
answers

86
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), ...

0
votes

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}

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 $n$...

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)| ...

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?

0
votes

1
answer

168
views

### 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: {\...

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.
...

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 ...

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 ...

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, ...

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)-...

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 ...

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)...

0
votes

0
answers

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(...

2
votes

1
answer

170
views

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

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

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 ...

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 $u(x,0)=g(x), \forall x\in \Bbb R$...

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_\...