Questions tagged [laplace-transform]
The laplace-transform tag has no usage guidance.
123
questions
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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
$$L^{-1}\left [ \left ( \frac{e^{as}-e^{-as}}{2} \right )F(s) \right]=\frac{1}{2}L^{-1}\left ( e^{as} F(s)\right )-\frac{1}{2}L^{-1}\left ( e^{-as}F(s) \...
- 59
1
vote
0
answers
129
views
Discretizing a differential operator which is a function of the derivative operator
Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$.
As an example, let $p(x)=x^{2}+2x$. Then ...
- 49
5
votes
2
answers
144
views
Limit of the extremal process of i.i.d. Gaussians see from the tip
I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
- 433
0
votes
1
answer
56
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Initial and final Theorem for upper and lower limits?
Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$.
Can we use upper limits or lower limits to ...
6
votes
1
answer
103
views
Convergence speed of the tail of distribution using Tauberian remainder theorem
This question may be related to this one.
Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem.
Let $f$ be some one-sided probability ...
- 205
-1
votes
1
answer
56
views
Finding the nth value of a dual series [closed]
I have a problem which i remember solving using Z transform in my uni. time, but i don't recall the EXACT way.
I have 2 series of numbers:
X[n] and Y[n]
X[0] = Y[0] = 1
X[n+1] = aX[n] + bY[n]
Y[n+1] = ...
- 107
0
votes
1
answer
71
views
Unique zero solution to a difference equation via Laplace transform
We want to prove that
the unique solution to the following difference equation is the null one:
$$
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}{...
- 525
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 ...
- 525
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0
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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*}
...
- 23
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 ...
- 23
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 ...
- 390
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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
$$
\mathcal{L}...
- 111
1
vote
1
answer
110
views
Inverse Laplace transform of $\frac{1}{s^a + 1}$ with $0 < a \leq 1$
Problem
I am looking for the following inverse Laplace transform,
$$
f(t) = \mathcal{L}^{-1}\left[\frac{1}{s^a + 1}\right]
\;\;\;\;\;
\text{with}
\;\;\;\;\;
0 < a \leq 1.
$$
What I understand
...
- 111
0
votes
1
answer
221
views
Is it possible to use the Laplace Transform to calculate eigenvalues?
The relationship of Eigenvalues with Gradient Descent
Let $A$ be a symmetric (and thus diagonalizable) matrix, with diagonalization
$$A=VDV^T.$$
Let us define the quadratic function
$$f(x) = x^T A x.$$...
- 121
8
votes
0
answers
386
views
Modern treatment of Delange's Tauberian Theorem
Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper:
H. Delange - Généralisation du théorème de ...
- 18.5k
2
votes
0
answers
181
views
What is the role of the Laplace transform in the topological recursion formalism?
While reading papers on topological recursion, among them The Laplace transform, mirror symmetry, and the topological recursion of Eynard–Orantin by M. Mulase, they describe the mirror symmetry ...
3
votes
0
answers
84
views
Inverse Laplace transform through contour integration
How can I prove that in formal way, this function doesn't have inverse Laplace transform.
$$
F(s)=\frac{\sin(s)}{\sqrt{s}}
$$
Strictly it should be in Bromwich contour method.
Could you please tell ...
- 31
2
votes
1
answer
611
views
Why is it possible to use the Inverse Laplace transform to get CDF?
I just saw the following on wikipedia about Laplace transformations:
"In probability theory and applied probability, the Laplace transform is defined as an expected value. If $X$ is a random ...
- 23
0
votes
1
answer
151
views
Laplace transform and Laguerre Polynomials
What is the kernel $K(t)$ of the following Laplace transform equation:
$$\int_{0}^{+\infty}e^{-(x+y)t} K(t) dt= \sum_{n=0}^{\infty}\varphi_{n}^{\alpha}(x)\varphi_{n}^{\alpha}(y),$$
where $\varphi_{n}^{...
- 159
1
vote
1
answer
678
views
What is the inverse Laplace transform of $\frac{(1/s)_{n}}{s} $?
Introduction
So far, I have found (p. 5) the following generating functions of the unsigned Stirling numbers of the first kind:
\begin{equation} \tag{1} \label{1} \sum_{l=1}^{n} |S_{1}(n,l)|z^{l} = (z)...
- 3,463
2
votes
0
answers
111
views
Regularization of the area under hyperbola
So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
- 8,368
1
vote
0
answers
52
views
Fractional power of the operator $\mathcal{L}_t[t f(t)](x)$ and equivalence of divergent integrals
I wonder whether an expression for fractional power of operator $\mathcal{L}_t[t f(t)](x)$ that involves Laplace transform can be derived?
I am asking this because this operator preserves the area ...
- 8,368
2
votes
2
answers
244
views
Laplace transform calculation
Please can someone help me? I have tried to find the Laplace transform of the form:
$$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$
...
- 159
3
votes
0
answers
99
views
Solving a integro-differential equation
I am trying to solve an integro-differential equation:
$$ {\frac{d}{dt}} f(t)=\int_0^t k(t-\tau)S(\tau)f(\tau) d\tau $$
with initial condition $f(0)=1$
If $k(t)=c\delta(t)$ with $c$ being constant, ...
- 71
1
vote
2
answers
213
views
Can we meaningfully ascribe values to these divergent integrals?
My gut feeling is that
$\int_0^\infty (1-\frac1{x^2})dx=0$
$\int_0^\infty (x-\frac2{x^3})dx=0$
$\int_0^\infty (x^2-\frac6{x^4})dx=0,$
etc, and in general,
$\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$
...
- 8,368
0
votes
1
answer
106
views
Can I express this random variable in terms of known distributions?
By computing the Laplace transform of the total length of a random tree (the nested Kingman coalescent tree with coalescence rates $\gamma$ for the individuals and $\gamma'$ for the species), we would ...
- 1
2
votes
1
answer
261
views
Is inverse Laplace Transform of a power of $s$ a positive function?
It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace ...
- 103
1
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1
answer
526
views
Inverse Laplace Transform using contour integration
So math stack exchange isn't really helping much with this.
So initially, I'm proving the inverse laplace transform using contour integration.
This is a good starting point for my research when I ...
- 19
1
vote
1
answer
53
views
Laplace transform of the product of two gammas
Suppose that $X$ and $Y$ are both gamma distributed with shapes $a,b$ and unit scales / unit rates. To fix ideas, X has Laplace transform given by:
$$L_X(t) = \mathbb{E}(e^{-tX}) = (1 + t)^{-a}$$
How ...
- 623
1
vote
0
answers
59
views
Laplace inversion with residue theorem doesn't satisfy IC of IVP
I have the following initial value problem
$$
\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 ...
- 623
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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), ...
- 187
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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}
- 327
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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 $n$...
- 4,827
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votes
2
answers
178
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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
$$\int_{-\infty}^\infty |\mathcal{L} f(\sigma+i t)| ...
- 16.7k
7
votes
1
answer
844
views
Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
- 3,343
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votes
1
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168
views
Is this integral transform related to the Laplace transform?
The Laplace transform of a function $f(t)$, defined for all real numbers $t \geq 0$, is the function $F(s)$, defined by
$${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}.$$
Let $\varphi: {\...
- 1
2
votes
1
answer
173
views
Integral transformation, Laplace-like
Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)?
$$
\int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt
$$
It resembles somewhat the Laplace transformation.
...
- 89
2
votes
1
answer
750
views
Bromwich integral transformed to an integral on the real axis
I am new in complex integration and inverse Laplace transforms. I already asked this question on math.se but got no answer.
The author of a textbook claims that the inverse Laplace transform has ...
- 2,053
2
votes
1
answer
190
views
Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization)
I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also ...
- 1,153
1
vote
0
answers
86
views
Is harmonic mean of linear functions a Bernstein function?
According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function:
$f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$
is a Bernstein function, ...
- 83
1
vote
0
answers
85
views
Expressing 1-e^{-z} as a Fourier integral
According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-...
- 83
3
votes
1
answer
150
views
Asymptotics for an exponential generating function from an ordinary
I'm interested in taking an ordinary generating function $$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!}.$$ I would then ...
- 131
0
votes
1
answer
220
views
Weak continuity under Laplace transform
Let the sequence $u_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $U_n(s)$ and $U(s)$?
$U_n(s)$ converges point-wise to $U(s)...
- 385
0
votes
0
answers
270
views
Laplace transform of a random variable: Inversion formula from an interval
Let $X$ be a non-negative random variable with a CDF $F$. Let $L_X(t)$ denote the Laplace transform of $F$, i.e.,
\begin{align}
L_X(t)=E[ e^{-tX}], \quad t \ge 0
\end{align}
It is known that $L_X(...
- 599
2
votes
1
answer
170
views
Can Mellin transform be applied in this function? What's the result?
$$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}$$
- 1,183
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votes
0
answers
119
views
Decay of Laplace (or Mellin) transform beyond region of convergence?
Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
- 16.7k
1
vote
0
answers
166
views
Diffusion equation solution using Laplace transform [closed]
Consider the operator
$$
L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t}
$$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...
- 111
1
vote
1
answer
85
views
Identity for stable Lévy subordinator
I want a proof or a reference for the identity
$$
\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_\...
- 213