# Questions tagged [laplace-transform]

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### Math calculators to find inverse laplace transform of slightly complicated equation [migrated]

I have found out the inverse laplace of $\log\frac{s+4}{s-4}$ which is $\frac{2 sinh(4t)}{t}$ But I do not know how to proceed. The multiplication by s property does not seem to hold good here as the ...
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### Asymptotics for an exponential generating function from an ordinary

I'm interested in taking an ordinary generating function $$F(x)=\sum_{n\geq 1}m_nx^n$$ and converting it to an exponential generating function $$M(x)=\sum_{n\geq 1}m_n\frac{x^n}{n!}.$$ I would then ...
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### Can Mellin transform be applied in this function? What's the result?

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
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### Decay of Laplace (or Mellin) transform beyond region of convergence?

Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
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### Approximate method to extract behavior of a Laplace transform in an intermediate region

In the theory of random walks, Tauberian type theorems are often applied to extract the small or large-time behavior from a difficult equation. For example, the Montroll-Weiss formula describing a ...
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### Diffusion equation solution using Laplace transform [closed]

Consider the operator $$L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t}$$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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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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### 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 $\tilde{\theta}(s)$, ...
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### 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, ...
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### 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 ...