The laplace-transform tag has no wiki summary.
1
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2answers
56 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 ...
0
votes
0answers
27 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 ...
6
votes
0answers
152 views
Laplace Transform in the context of Gelfand/Pontryagin
Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose $L^1$ ...
0
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1answer
98 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
65 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)
...
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0answers
64 views
Having problems with solving Lindley's equation in G/G/1 queuing?
Lindley's integral equation is as follows
$$W(y)=\int_{u=-\infty}^{y}W(y-u)dC(u),$$for $y\ge0$;
and
$$W^{-}(y)=\int_{u=-\infty}^{y}W(y-u)dC(u)$$,for $y<0$.
So we have
...
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0answers
41 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 ...
1
vote
2answers
89 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
137 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
166 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 ...
14
votes
1answer
401 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 $Mf(\sigma-it)$ ...
2
votes
3answers
945 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}
...
0
votes
0answers
147 views
Inverse Laplace transform of the function: $F(s)=e^{-a\sqrt{s(s+b)}}$
I would like to find inverse Laplace transform of the function:
$$F(s)=e^{-a\sqrt{s(s+b)}}$$
which $a$ and $b$ are positive real numbers and $s$ is a complex variable. It would be appreciated if ...
4
votes
1answer
138 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 ...
8
votes
0answers
145 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
274 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) \: ...
3
votes
0answers
90 views
Lower bounds of laplace transform of characteristic functions
Cross-posted on maths.stackexchange
I have the following integral:
\begin{equation}
f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt
\end{equation}
where $\varphi_X(t)$ is the characteristic function ...
1
vote
1answer
205 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
169 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
185 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
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0answers
649 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
509 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 ...
4
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
207 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 ...
23
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
148 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
294 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 ...
5
votes
3answers
977 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) ...
2
votes
0answers
268 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 ...
12
votes
2answers
910 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
494 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 ...
9
votes
6answers
30k 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
521 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
501 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 ...
16
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6answers
4k 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 ...
33
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9answers
6k 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 $f(t)$, ...
