Questions tagged [laplace-transform]
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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 $f(t)$, ...
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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 ...
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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 ...
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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 ...
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Motivation and physical interpretation of the Laplace transform
Concerning the one-sided Laplace transform,
$$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$
what is a motivation to come up with that formula? I am particularly interested in "physical&...
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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 $Mf(\sigma-it)$ ...
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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:
$$f(x) =...
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Laplace Transform in the context of Gelfand/Pontryagin
Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...
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Proof of complete monotonicity of a binomial function
By plotting the function and its derivatives, one can easily be convinced that the function
$$f(x):=\log\binom{x}{p x}=\log\Gamma(x+1)-\log\Gamma(px+1)-\log\Gamma((1-p)x+1),$$ defined for $x>0$ and ...
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Getting a differential equation for a function from a functional equation of its Mellin transform
If $f$ is a locally integrable function then its Mellin transform
$\mathcal{M}[f]$ is defined by
$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$
This integral usually converges in a ...
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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 ...
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What is the intuition behind applying the Mellin Transform to prime distribution?
I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem.
I understand that applying the Mellin Transform to the partial sum of the van ...
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Modern treatment of Delange's Tauberian Theorem
Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper:
H. Delange - Généralisation du théorème de ...
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Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
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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}
...
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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 $X(...
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Growth of (integral of) Laplace transform of a function of compact support as $Re \to -\infty$
Let $f:[0,\infty)\to \mathbb{R}$ be supported on $[0,1]$, with $\int_0^1 f(x) dx = 1$. Let $\mathcal{L} f$ be its Laplace transform. How slowly may
$$\int_{-\infty}^\infty |\mathcal{L} f(\sigma+i t)| ...
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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 ...
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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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On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
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Convergence speed of the tail of distribution using Tauberian remainder theorem
This question may be related to this one.
Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem.
Let $f$ be some one-sided probability ...
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Limit of the extremal process of i.i.d. Gaussians see from the tip
I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
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How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?
I tried to find an inverse Laplace transform by series as follows
$$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$
and by ...
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Why is it possible to use the Inverse Laplace transform to get CDF?
I just saw the following on wikipedia about Laplace transformations:
"In probability theory and applied probability, the Laplace transform is defined as an expected value. If $X$ is a random ...
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Long tail property of Laplace transforms
A function $F: \mathbb R_+ \rightarrow \mathbb R_+$ is said to be long tailed if $F(\infty)=0$ and for all $y \geq 0$ $$\frac{F(x+y)}{F(x)} \rightarrow 1, \quad x\rightarrow \infty.$$
Let $\mu$ be a ...
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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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Upper bound for an inverse Laplace transform
Can anyone see how to get a tight upper bound for the function defined in terms of the inverse Laplace transform below?
$$g_\text{sgd}(t)=\mathcal{L}^{-1}\left[\left(\frac{s}{2}+\frac{5 \sqrt{\frac{2}{...
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Hardy–Littlewood Tauberian theorem for Laplace transform
The Hardy–Littlewood Tauberian theorem for Laplace transform in Chapter XIII in "An Introduction to Probability Theory and Its Applications" by Feller reads as follows
Let $F : [0,\infty) \...
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Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
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Paley–Wiener theorem for functions with exponential decay
I feel like this should be well-known, but haven't been able to find any reference so far. Consider the set of all smooth functions on $\mathbb{R}$ such that $$\sup_{x\in \mathbb{R}} |e^{\alpha x} f^{(...
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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 ...
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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 $(\...
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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 $\int_0^\infty f(t) \exp(-xt^2) \: ...
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$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$
I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW ...
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Is it possible to use the Laplace Transform to calculate eigenvalues?
The relationship of Eigenvalues with Gradient Descent
Let $A$ be a symmetric (and thus diagonalizable) matrix, with diagonalization
$$A=VDV^T.$$
Let us define the quadratic function
$$f(x) = x^T A x.$$...
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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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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 +...
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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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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 ...
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Smoothness and decay correspondence for Laplace transform
For the Fourier transform, there are various theorems formalizing a correspondence between the smoothness of a function and the rate of decay of its Fourier transform. For example, if a function is $n$...
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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 $, ...
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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 ...
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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 ...
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Integral of a function changing sign
By some numerical tests, we can see that the following function is negative on $(0,1)$:
$$\small f(x)=\int_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-...
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Does the hitting time of +1/-1 of a Brownian motion posess a density?
The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$.
I define $T=\inf \{t>0,|W|(t)= 1\}$. ...
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What is this equation, written on a wall? [closed]
I was asked to ID the following, but can't figure out what it's for. Laplace Transform of acceleration (x double-dot)?
(Sorry that I can't provide a sharper image - this is all I have access to)
I ...
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Is inverse Laplace Transform of a power of $s$ a positive function?
It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace ...
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Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization)
I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also ...
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Inverse Laplace transform of $sech(\sqrt{2\lambda})$ and Brownian motion occupation time
In dimension 3 we have that for $T=\int_{[0,\infty)}1_{B_{t}\in B(0,1)}dt$ has the Laplace transform
$$E[e^{-\lambda T}]=sech(\sqrt{2\lambda}).$$
And in dimension 1 we have the same for $\tau=\min\{t:...
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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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