Questions tagged [laplace-transform]

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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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1answer
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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1answer
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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1answer
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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0answers
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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0answers
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)-...
3
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1answer
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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1answer
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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1answer
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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0answers
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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1answer
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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1answer
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+...
3
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1answer
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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1answer
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 ...
4
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1answer
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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1answer
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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1answer
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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1answer
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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2answers
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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1answer
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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3answers
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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0answers
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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0answers
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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1answer
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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1answer
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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1answer
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 ...
2
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1answer
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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0answers
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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2answers
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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2answers
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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0answers
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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0answers
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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2answers
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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2answers
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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1answer
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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0answers
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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0answers
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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2answers
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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1answer
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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1answer
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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0answers
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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0answers
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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2answers
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 ...