# Questions tagged [laplace-transform]

The laplace-transform tag has no usage guidance.

105
questions

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### 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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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)...

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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 ...

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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 ...

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202 views

### Laplace transform calculation

Please can someone help me? I have tried to find the Laplace transform of the form:
$$\int_{0}^{+\infty} (v+1)^{\nu}(2v+1)^{k}v^{\alpha} \exp(-pv), \mbox{ where }\alpha,\nu, k \mbox{ are integers }$$
...

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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, ...

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

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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 ...

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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 ...

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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 ...

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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 ...

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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
$$
\...

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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 ...

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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 ...

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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), ...

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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}

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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 $n$...

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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)| ...

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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?

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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: {\...

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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.
...

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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 ...

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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 ...

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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, ...

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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)-...

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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 ...

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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)...

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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(...

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167 views

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

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

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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 ...

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

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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_\...

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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 ...

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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+...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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) ...

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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 $T=\inf \{t>0,|W|(t)= 1\}$. ...

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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 ...

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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 ...

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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}...

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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 ...

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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 ...

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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 ...

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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^{(...

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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 ...