# Questions tagged [laplace-transform]

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88
questions

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### Math calculators to find inverse laplace transform of slightly complicated equation [migrated]

I have found out the inverse laplace of $\log\frac{s+4}{s-4}$ which is $\frac{2 sinh(4t)}{t}$
But I do not know how to proceed. The multiplication by s property does not seem to hold good here as the ...

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91 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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63 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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198 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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**1**answer

101 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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70 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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74 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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80 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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122 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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111 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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160 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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101 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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18 views

### Approximate method to extract behavior of a Laplace transform in an intermediate region

In the theory of random walks, Tauberian type theorems are often applied to extract the small or large-time behavior from a difficult equation. For example, the Montroll-Weiss formula describing a ...

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107 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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66 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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115 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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105 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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57 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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132 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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152 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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147 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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215 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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**1**answer

101 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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301 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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595 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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762 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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81 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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97 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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102 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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617 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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487 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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368 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 ...

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199 views

### Upper bounds on the inverse Laplace transform

We define the Laplace transform of a non-negative function $f : \mathbb{R_+} \to \mathbb{R_+}$ by
$$\mathcal{L}f(q) \triangleq \int_0^{+\infty}f(t)e^{-qt}dt,$$
where $q$ is in the domain of ...

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101 views

### Laplace transform of function that oscillates n times

I am interesting in understanding the following idea:
Suppose we have a function $f(x)$ such that $f \in L^1([0,\infty)), |f(x)| \leq C \exp(-\rho x)$, $\int_0^\infty f > 0$. Further, suppose $f$ ...

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922 views

### Integro-differential equation

I have an equation of the type $$f-\frac{\sigma^2} 2 \frac{d^2 f}{dx^2}-\frac{df}{dx} = \int_0^\infty \left(\frac{df}{dx}\right)^2\exp(ax+f(x)) \, dx.$$
It is an integro-differential equation but the ...

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396 views

### Inverse Laplace transform of $sech(\sqrt{2\lambda})$ and Brownian motion occupation time

In dimension 3 we have that for $T=\int_{[0,\infty)}1_{B_{t}\in B(0,1)}dt$ has the Laplace transform
$$E[e^{-\lambda T}]=sech(\sqrt{2\lambda}).$$
And in dimension 1 we have the same for $\tau=\min\{t:...

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82 views

### Looking for modern reference for asymptotic of Barnes integral

I am reading a paper (arXiv:1404.6407, by Galkin, Golyshev and Iritani) where the authors need to use the statement that for $z\to 0+$ (and in fact
in a sector) the integral
$$
\int_{1+\rm i \mathbb R}...

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66 views

### Compute the following Laplace transform [closed]

I'm trying to Laplace transform the function
\begin{equation}
|\theta(t)|\sin(l\theta(t)),
\end{equation}
where $\theta(t)$ is any function of t.
I want to express the result with $\tilde{\theta}(s)$, ...

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781 views

### Solving a general, constant-coefficient, first-order, two-indep-variable system of PDEs

I have the following system of PDEs that I want to solve as "analytically" as possible:
$$\left(\partial_t + A\partial_x + B\right)\mathbf{u}(t, x) = 0,$$
where $A$ and $B$ are constant, ...

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255 views

### How to find the Inverse Laplace Transform of the following?

I have a Laplace tranform in the form given below
$\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$
Can some one help me to find ...

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377 views

### Inverse Laplace transform of a hypergeometric function

This is a repost from Math Stack-exchange where I did not manage to get an answer.
https://math.stackexchange.com/questions/1491027/inverse-laplace-transform-of-a-hypergeometric-function
I managed ...

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169 views

### Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform,
$$
f(s)=\int_0^\infty e^...

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96 views

### Inverse Laplace Transform involving irrational powers

Could anybody please suggest a reference or a possible solution how to invert the Laplace transform of
$e^{-(s^{\alpha}+\lambda)^{\beta}}$, where $0<\alpha<1$ and $0<\beta<1$.

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201 views

### Why is taking the inverse Laplace transform valid in this case?

Assume $F \in L^2([0,\infty))$, so that the Laplace-transform $L[F]$ is well-defined. Assume furthermore, that
$$
y \mapsto \frac{L[F](iy)}{1+L[F](iy)}
$$
is in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$, ...

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191 views

### inverse Laplace transform of the determinant

what is the inverse Laplace transform of the function
$x\mapsto\det x$? (where $x\in\mathbb{R}^{n\times n}$ is a symmetric metric, if necessary at all.)

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437 views

### Variations on the Mellin and Dirichlet transforms

There are a number of variations on the Laplace transform that turn up all over math. Some examples:
$\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
$\sum_{-\infty}^{\infty} f(t)z^{-...

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85 views

### Functions whose Laplace transforms have prescribed behavior at minus infinity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non-negative function with entire Laplace transform $\hat{f}$ (in particular $\lim_{t\to \infty}e^{st}f(t)=0$ for all $s$), and $p_0$ a positive integer. ...

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51 views

### How to find Laplace Transform of fractional-order differential systems

Let us consider the fractional-order complex-valued dynamic system as
$D^\alpha z(t)=-Az(t)+Bz(t-\tau),\ z\in \mathbb{C}$.
How to find the characteristic equation and the Laplace transformation?

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229 views

### compute the limit of a rational function

Suppose I have a rational function defined by ($s$ complex)
$$
f(s)=w^T s(sI-Q)^{-1} v
$$
for nonzero column vectors $w,v$ and a (large) square matrix $Q$. Further assume that $Q$ is singular and that ...