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-4
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0answers
30 views

Finding Laplace inverse transformation of a product series [on hold]

compute the inverse Laplace transformation of the following equation. \begin{align*} f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\ &=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}. \end{align*}...
3
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0answers
57 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+...
0
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0answers
76 views

Concerning some Tauberian-type asymptotics of Laplace transform involving $e^{-\sqrt{s}}$

There are some well-known Tauberian theorems concerning the asymptotics of the original function (say as $t$ tends to $0$) and that of its Laplace transform (as $s$ tends to infinity). I want to ask a ...
0
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0answers
35 views

Initial value theorem for Fourier transform

Initial value theorem states that for a bounded function $f(t) = O(e^{ct})$ and an existing initial value, one-sided Laplace transform $F(s) = \int\limits^\infty_{0^-}f(\tau)e^{-s\tau}\mathrm{d\tau}$ ...
3
votes
1answer
97 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 ...
1
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0answers
51 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
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1answer
82 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
111 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
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1answer
136 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
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1answer
95 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
53 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
273 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
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1answer
249 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
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3answers
646 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
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0answers
38 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
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0answers
45 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
94 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
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1answer
603 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 ...
3
votes
1answer
215 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
298 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
votes
1answer
142 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
98 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 $f$ ...
1
vote
2answers
711 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 ...
3
votes
2answers
226 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:...
0
votes
0answers
45 views

DIfference between transforms,

I have a question regarding the difference between Laplace transform and so called Carson-Laplace transform. I mean what's the motivation behind the second one, why was it invented? Is there some area ...
2
votes
0answers
74 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}...
1
vote
0answers
62 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 $\tilde{\theta}(s)$, ...
0
votes
0answers
731 views

Inverse Laplace transform of matrix exponential

I have the following Laplace-transformed, matrix-valued function: $$U(s) = e^{As + B},$$ where $A$ and $B$ are diagonalizable, noncommuting (but very close to commuting, if that's useful -- $B$ is ...
1
vote
2answers
547 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, ...
1
vote
2answers
169 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 ...
3
votes
1answer
254 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 ...
1
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0answers
130 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^...
1
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0answers
80 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 $0<\beta<1$.
4
votes
2answers
145 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 $L^2(\mathbb{R})\cap L^1(\mathbb{R})$, ...
1
vote
1answer
167 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.)
5
votes
1answer
361 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^{-...
3
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0answers
82 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. ...
1
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0answers
46 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?
1
vote
2answers
226 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 ...
2
votes
0answers
185 views

Existence of zero-free strip of a Laplace transform (edited ..)

Problem Let $\beta$ be a probability measure on $\mathbb{R}$, and define $$ K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is well-...
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0answers
160 views

Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as $$ \mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx $$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
4
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0answers
902 views

Parseval's theorem

In operational calculus there is Parseval's theorem, which states that if $ f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p) $ and both $ F(p) $ and $ \Phi(p) $ are analytical in $ Re p \geq 0 $, ...
2
votes
2answers
806 views

Sum of two independent random variables

Let $\xi, \eta, \eta'$ be non-negative random variables such that: $\eta \stackrel{\mathcal{L}}{=} \eta'$, $\xi + \eta \stackrel{\mathcal{L}}{=} \xi + \eta'$, $\xi$ and $\eta'$ are independent. Does ...
1
vote
2answers
447 views

a limit of the laplace transform and its derivative

If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
3
votes
1answer
267 views

Integral representation of the resolvent of a semigroup

Let $T(t)$ be a $C_{0}$-semigroup with the generator $A$. Now, does the so called integral representation of the resolvent $$ (\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt $$ hold for ...
14
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0answers
581 views

Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...
0
votes
1answer
132 views

Extracting moments from a special Z-transform

Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that \begin{equation}\frac{X_n-n\mu}{\sqrt{n}\sigma}...
1
vote
1answer
2k views

Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) \begin{...
2
votes
2answers
322 views

How do I estimate/bound the error in an inverse Laplace transform?

Suppose I have a Laplace transform, $$ F(s) = \int_0^{\infty}dx\ f(x)e^{-s x} \ . $$ I know that $$ F(s) \approx e^{A/(4s)} $$ (for $s$ real) where $A$ is very large, and I want to estimate $f(x)$...
1
vote
1answer
360 views

An Integral Functional Equation

Let $f$ be a non-negative function supported and integrable on the positive real axis, such that $$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$ where $c[p]$ a number (functional) dependent on function $...