The laplace-transform tag has no usage guidance.

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### 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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### 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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81 views

### Inverse Laplace Transforms of Exponential Form

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...

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106 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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215 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} ...

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### 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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### 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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214 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 ...

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131 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 ...

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105 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 $f(x-x_0)$? ...

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397 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 $, ...

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568 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.
...

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112 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 ...

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145 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 ...

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386 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 ...

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**1**answer

110 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
...

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366 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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241 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 ...

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232 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 ...

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257 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 ...

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527 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 $Mf(\sigma-it)$ ...

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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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189 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 ...

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181 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 ...

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300 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) \: ...

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319 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 ...

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### 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 ...

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201 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 ...

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### 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 ...

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772 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 ...

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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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245 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 ...

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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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169 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 ...

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336 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 ...

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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) ...

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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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### 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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### 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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### 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 ...

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53k 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 ...

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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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### 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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### 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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### 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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### 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 $f(t)$, ...