The tag has no wiki summary.

learn more… | top users | synonyms

-4
votes
0answers
36 views

Inverse Laplace Transformation [closed]

I have a question about laplace transformation. $\frac{8s+4}{s^2+23}$ I tried to split them. $\frac{8s}{s^2+23}$ is similar to cos and $\frac{4}{s^2+23}$ is similar to sin. Can anyone help? ...
1
vote
0answers
106 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 ...
0
votes
0answers
88 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 $f(x-x_0)$? ...
2
votes
0answers
201 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
500 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. ...
1
vote
2answers
90 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
109 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 ...
9
votes
0answers
275 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
103 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 ...
0
votes
1answer
185 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) ...
1
vote
0answers
53 views

Approximate Laplace (forward) transforms

Given f(t) let's write Lf(s) = INT_0^inf f(t) exp(-ts)dt for its Laplace transform. I lately find myself looking at expansions of the form SUM_i a_i exp(-t_i s) that converge to Lf of some given ...
2
votes
2answers
178 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 ...
0
votes
1answer
173 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 ...
4
votes
1answer
203 views

Characterization of the Laplace Transform

One of the main properties of the Laplace transform is given by the convolution theorem. $$\mathcal{L}(f*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$ Question: Is there a full characterization of the ...
16
votes
1answer
466 views

Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
4
votes
4answers
2k views

Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} ...
4
votes
1answer
156 views

Relations involving Stirling numbers of second kind

While inverting a Laplace transform using Post's inversion formula I found the following expression: $$ \sum_{k=1}^n S^n_k \ x^k(\alpha)_k $$ where $S^n_k$ is a Stirling number of second kind and ...
7
votes
0answers
159 views

Inferring asymptotic behaviour from the dominant pole of the Laplace transform

Hi, I am reposting the following question with the hope that a more detailed description will lead to a more descriptive response: dominant pole in the laplace transform I have a vector function ...
3
votes
1answer
286 views

Using a quadratic kernel instead of a linear kernel in the Laplace transform

Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: ...
1
vote
1answer
258 views

dominant pole in the laplace transform

hi, I have a function $X(t)$ whose Laplace transform $\hat{X}(s)$ has a unique pole of largest real part $x_0$, which is a real number. I am able to show that for each $t$, $X(t)$ is a convergent ...
1
vote
0answers
202 views

Wiener-Hopf Integral/Lindley's Equation

Lindley's equation is well known within queueing theory and is as follows $F(y) = - \int_0^\infty F(x)dH(y-x)$ However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which ...
4
votes
2answers
193 views

On a certain generalization of the Laplace transform

Let $\alpha$ be a positive constant, $\mu$ be a Borel nonnegative measure in $\mathbb{R}^n_+$. We can define a transform $$ \tilde{L}\[\mu\](p) = \int\limits_{\mathbb{R}^n_+} e^{-(p_1 x_1 + \ldots ...
0
votes
0answers
978 views

An inverse Laplace transform involving Error function

Dear MOs, I need to calculate the inverse Laplace transform of the following function $$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$ I have checked, among many ...
0
votes
2answers
624 views

Understanding the inverse Laplace transform of a function with essential singularities

I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in ...
3
votes
2answers
2k views

How does the Laplace Transform work for circuit analysis? [closed]

I would like to understand how signals transformed from the time domain to the frequency domain for algebraic manipulation, can be transformed back to give solutions in the time domain. Knowing how to ...
1
vote
0answers
226 views

On the generalisation of the Laplace transform

I consider a measure transform $A$ given by $$ A\mu(x) = \int\limits_{\mathbb{R}^n_{+}} e^{-g(x,y)} \mu(dy) $$ where $g(x,y)$ is some positive smooth function, $\mu$ is a Borel measure. Is it a ...
25
votes
6answers
4k views

What does Mellin inversion “really mean”?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
2
votes
1answer
156 views

Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution

I am trying to calculate the Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution. I am ...
3
votes
0answers
314 views

Laplace transform of a stopping time for stochastic volatility models

Let $V_t$ be a solution of the SDE $$dV_t=V_t(rdt+\sigma_t dW_t) $$ where $\sigma_t$ satisfies some other SDE $$d\sigma_t=\alpha(t,\sigma_t)dt+\beta(t,\sigma_t)dW^{\\ \prime}_t $$ and $W_t$ and ...
8
votes
3answers
1k views

When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as: $f(x) ...
3
votes
0answers
288 views

inverse Laplace transform of $\delta_1(\cdot)$

Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...
13
votes
2answers
964 views

Getting a differential equation for a function from a functional equation of its Mellin transform

If $f$ is a locally integrable function then its Mellin transform $\mathcal{M}[f]$ is defined by $$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$ This integral usually converges in a ...
2
votes
2answers
2k views

Laplacian operator and relation to the Laplace Transform

I'm trying to understand why the Laplacian operator is used in blob detection in image analysis. I must admit that in trying to figure out why the Laplacian is useful in this application, I've really ...
2
votes
1answer
513 views

method of moments and Laplace transform from Shepp and Lloyd

Again from the Shepp and Lloyd paper "ordered cycle lengths in a random permutation", I found this puzzling equality. This one might require access to the paper itself since it's quite a mouthful: In ...
11
votes
6answers
39k views

Fourier vs Laplace transforms

In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...
2
votes
2answers
612 views

Can we extract information about how fast a function decay from its Laplace transform?

My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform. More concrete case, let $f:\mathbb{R} ...
6
votes
5answers
4k views

Applied mathematics Books (graduate level)

What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems? There is a lot of books on ...
3
votes
2answers
515 views

Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ansätze do you know ...
22
votes
6answers
5k views

Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...
39
votes
9answers
7k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...