Questions tagged [laplace-transform]
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105
questions
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1answer
61 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
1answer
164 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
0answers
68 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
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0answers
24 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
2answers
202 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
0answers
81 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
2answers
185 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
1answer
76 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
1answer
106 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
1answer
255 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
1answer
40 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
0answers
47 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
$$
\...
0
votes
0answers
36 views
Laplace transform of sum of random variables in first hitting time problem
Let me refer to the example here.
Suppose $X$ is a birth-death (BD) process (represents population size) that evolves by:
$X \to X+1$ if a birth occurs with rate $\mu$,
$X \to X-1$ if a death occurs ...
0
votes
1answer
193 views
Laplace transform inversion
I have a probability distribution that is defined through it's Laplace transform by :
$$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$
Using R and the invLT package, i have a numerical ...
1
vote
0answers
67 views
About the computation of the inverse Laplace transform [closed]
I have several questions about the inverse Laplace transform:
If $F(s)$ is a smooth real-valued function vanishing on a large subset of $\mathbb{R}$ (e.g. $F(s)$ is supported on a bounded interval), ...
0
votes
1answer
47 views
Laplace transforms of fractional equation
is there a finite expression of the Laplace transforms of the function
\begin{align}
L\left[ {\frac{{{x^n}}}{{{{(1 + x)}^m}}}} \right]\quad n \ge m
\end{align}
3
votes
0answers
94 views
Smoothness and decay correspondence for Laplace transform
For the Fourier transform, there are various theorems formalizing a correspondence between the smoothness of a function and the rate of decay of its Fourier transform. For example, if a function is $n$...
6
votes
2answers
150 views
Growth of (integral of) Laplace transform of a function of compact support as $Re \to -\infty$
Let $f:[0,\infty)\to \mathbb{R}$ be supported on $[0,1]$, with $\int_0^1 f(x) dx = 1$. Let $\mathcal{L} f$ be its Laplace transform. How slowly may
$$\int_{-\infty}^\infty |\mathcal{L} f(\sigma+i t)| ...
7
votes
1answer
757 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
1answer
101 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
1answer
169 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.
...
0
votes
1answer
495 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
1answer
133 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
0answers
84 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
0answers
81 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
1answer
105 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
1answer
166 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
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0answers
155 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
1answer
167 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
0answers
116 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
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0answers
120 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
1answer
72 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_\...
1
vote
1answer
150 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 ...
3
votes
0answers
65 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+...
3
votes
1answer
106 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 ...
2
votes
0answers
59 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 ...
0
votes
1answer
148 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 ...
4
votes
1answer
177 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 ...
1
vote
1answer
151 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 ...
1
vote
1answer
288 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 ...
1
vote
1answer
105 views
representation of the Wright function
The two-parameter Wright function http://dlmf.nist.gov/10.46 is defined as the infinite series
$$
\phi (\alpha, \beta \, | z)=\sum\limits_{k=0}^\infty \frac{z^k}{\Gamma(k+1) \Gamma(\alpha k + \beta) ...
3
votes
2answers
320 views
Does the hitting time of +1/-1 of a Brownian motion posess a density?
The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$.
I define $T=\inf \{t>0,|W|(t)= 1\}$. ...
1
vote
1answer
750 views
Inverse Laplace transform to get CDF
I have the following problem. If I can get some help, I would greatly appreciate it. I am trying to replicate a particular research paper and came across a problem:
Suppose $X$ is a birth death ...
15
votes
3answers
804 views
Proof of complete monotonicity of a binomial function
By plotting the function and its derivatives, one can easily be convinced that the function
$$f(x):=\log\binom{x}{p x}=\log\Gamma(x+1)-\log\Gamma(px+1)-\log\Gamma((1-p)x+1),$$ defined for $x>0$ and ...
1
vote
0answers
86 views
How to solve differential equation for cylindrical diffusion?
How the differential equation for diffusion along a hollow cylinder,
$$ \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}...
2
votes
0answers
133 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 ...
1
vote
0answers
104 views
Existence of a Laplace transform that takes specific values on the integers
The classical Marcinkiewicz theorem (1939) states that if a random variable $X$ has a Laplace transform/characteristic function of the form $\mathbb{E}(e^{tX})=e^{P(t)} $ with $P$ a polynomial, then ...
2
votes
1answer
635 views
What is this equation, written on a wall? [closed]
I was asked to ID the following, but can't figure out what it's for. Laplace Transform of acceleration (x double-dot)?
(Sorry that I can't provide a sharper image - this is all I have access to)
I ...
4
votes
1answer
579 views
Paley–Wiener theorem for functions with exponential decay
I feel like this should be well-known, but haven't been able to find any reference so far. Consider the set of all smooth functions on $\mathbb{R}$ such that $$\sup_{x\in \mathbb{R}} |e^{\alpha x} f^{(...
5
votes
1answer
410 views
Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$
EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression.
I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
