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Questions tagged [laplace-method]

Laplace's method is a way of approximating integrals and related quantities, like expectations, see https://en.wikipedia.org/wiki/Laplace%27s_method

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How to find the asymptotics of this integral

Let $m,n,l_0$, $k_j$, $j=1,\cdots,l_0$ be positive integers, and they satisfy $$k_1+ \cdots + k_{l_0}=n\quad\mbox{and}\quad m\ll n^{\frac{1}{4}}.$$ Consider $$x=(\widetilde{x_1},\cdots,\widetilde{x_n})...
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Solving the PDE through Laplace Transform Method

I have a particular PDE as shown below: $$ \frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial x^2} + xe^{-\gamma x} $$ with Boundary conditions as shown, $$ \nu, \gamma >0 ~~~~1)~u(0, ...
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Asymptotic formula of a definite double integral

Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$ ...
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Asymptotic expansion of $\int_0^1 e^{x(tp-\frac{1}{2}t^2)}dt$ as $x\to\infty$ for different $p$

Let $p\in \mathbb{R}$, I would like to investigate the asymptotic behavior of the following integral: $$\int_0^1 e^{x(tp-\frac{1}{2}t^2)}dt$$ as $x\to\infty$. In particular, I would like to know how ...
John's user avatar
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Convergence rate of Laplace principle in Large Deviations

Let $H:A \to \mathbb R$ be a continuous function defined on a compact subset $A\subset \mathbf{R}^n$. Then the Laplace principle shows that $$ \lim_{\theta\to \infty }\frac{1}{\theta}\log \int_A ...
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Has anybody ever heard about this theorem? $L\{f'(t)\}(0)+f(0)=k$

Does anybody know the following theorem? $y(0)+f(0)=k$. Let $f:D\rightarrow C$ be a Laplace function with additional constant $k$, and $y(s)=L\{f'(t)\}$. If $y(0)$ is defined then: $y(0)+f(0)=k$.
Alessandro Pini's user avatar
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Concentration of Gibbs measures with converging energy functions

Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$ and let $Q$ be a fixed probability measure on $A$. For each $\theta>0$, define the probability ...
John's user avatar
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Explicit bound on $\int_0^\infty t^{-(u+1/u)} du$?

I would like to give an explicit bound on $\int_0^\infty t^{u+1/u} u^k du$. It's easy to reduce to $k=0,1$ by integration by parts, and I take one can reduce matters to the case $k=0$ alone by ...
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Laplace’s method for asymptotic series expansion [closed]

I have this integral for which I need to calculate the leading asymptotic behavior: $$ I\left(x\right)=\int_0^{\frac{\pi}{2}}e^{-x\tan\left(t\right)}dt,\quad x\rightarrow \infty$$ $\phi\left(t\right)=\...
user1269414's user avatar
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to find inverse laplace transform of $f(s)=s\ln\left|\frac s{\sqrt{s^2+1}}\right|$

I tried the differential property of the Laplace transform to the logarithm component, then I tried the “multiplying by s” property, but the answer would be infinity. What do you think is the solution?...
Mohammed Mohammed's user avatar
5 votes
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Asymptotic expansion of an inverse Fourier integral

To find the $t\rightarrow +\infty$ limit of the following integral: $$ G(t)=\int_{-\infty}^{+\infty} \left(\frac{\sqrt{\frac{1}{1- 2 i \omega}}(3 i \omega -1)\omega^2}{(i \omega+1)(1-i \omega)^2}-\...
Navid's user avatar
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Laplace's method on $\int_0^\infty x^m\exp \left( - \frac{(x - \mu)^2}{2 \sigma^2} \right)\ dx$

I was looking to approximate this integral using ideas from Laplace's method, however I ended up with a $\color{red}{\text{divergent improper integral}}$. I want to know why such a method doesn't work ...
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Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals?

Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals? In particular, I am interested in asymptotic behaviour of the Riemann-Stieltjes ...
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Asymptotic equality of an integral

I need an asymptotic equality for the integral $$\int_{(0,1)^5} \left(\frac{x(1-x)y(1-y)u(1-u)v(1-v)w(1-w)}{1-(1-xyuv)w}\right)^n \frac{dxdydudvdw}{1-(1-xyuv)w} $$ where $$\int_{(0,1)^5}=\int_{0}^1\...
Max's user avatar
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Leading-Order Asymptotic Approximation of $\int^1_0 e^{-xt^2}\sin (t) dt$ as $x \rightarrow \infty$

I wanted to know if I did this calculation correctly. I used the Laplace Method and considered where in the range of integration we have the most contribution to the integral from the integrand. Here ...
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How to approximate this integral

I am trying to approximate an integral of the following form $$ F_n = \int_{-1}^1(1-u^2)^{(n-3)/2}e^{Anu}du $$ (I am trying to work through this paper....https://ieeexplore.ieee.org/document/6875199) ...
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Laplace's method: multivariate case with error term

The $d$-dimensional case of Laplace's method is usually given as something like $$ \int_D h(\mathbf{x})e^{-Mf(\mathbf{x})}d\mathbf{x} \sim \Big(\frac{2\pi}{M}\Big)^{d/2} \frac{h(\mathbf{x_0})e^{-Mf(\...
WithinCellsInterlinked's user avatar
12 votes
3 answers
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Leading order asymptotic behaviour of the integral $\int^1_0 \cos(xt^3)\tan(t)dt$

I'm trying to get the leading order asymptotic behaviour of the integral: $$\int^1_0 \cos(xt^3)\tan(t)dt$$ I'm trying to use the Generalised Fourier Integrals and the Stationary Phase Method, but I ...
bsaoptima's user avatar
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Using Integration by Parts to find leading order approximation of exponential integral

I'm trying to understand how finding leading order approximation of exponential integrals. Here is my integral: $$\int^{+\infty}_0 e^{-xt}\ln(1+t^2)dt$$ I need to use Integration by parts to then find ...
bsaoptima's user avatar
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Asymptotic behavior of a Fourier integral

Let $a \in \mathbb{R}_0, b>0$, then consider the integral $$I(t) = \int_{-\infty}^\infty d\omega e^{i\omega t}\frac{1}{ib|\omega|^a - \omega},$$ with $t>0$. I would like to understand the ...
Audrique's user avatar
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What is the general solution ( without start terms ) of laplace equation - $u_{ss}=-u_{tt}$

I had a question is Analysis fourier, which was: let $u(x,y)$ such that $u_{xx}-2u_{xy}+5u_{yy}=0$ when $s=x+y$ and $t=2x$ I solved it, and reached the conclusion of the Laplace equation ( it is true, ...
LearningToCode's user avatar
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2 answers
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On the asymptotics of $\int_0^\infty \exp\big(\!-\!\lambda e^x/x\big)\,dx$ [closed]

The question is about the asymptotics of $$I(\lambda):=\int_0^\infty \exp\big(\!-\!\lambda e^x/x\big)\,dx$$ as $\lambda\to\infty$. I ask this question here because this is a non-standard form of an ...
Diffusion's user avatar
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How can I improve my proof of Stirling's Theorem?

I'm trying to prove Robbin's inequality: $$ n! \le \sqrt{2 \pi n}(n/e)^n e^{1/(12n)}. $$ Step 1: I start from the integral formulation \begin{align} n! = \int_0^\infty x^n e^{-x} dx &= (n/e)^n\...
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Asymptotic expansion (Laplace Method) when Taylor series does not exist

So I've been struggling on calculating the asymptotic behaviour of the integral $$I(\alpha) = \int_0^\infty e^{-n(t+t^\alpha)}dt$$ as $n\rightarrow\infty$ and where $\alpha$ is a real number and $\...
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What is the meaning and intuition behind Laplace's method for the maximum of a function?

I am reading the book Bandit Algorithms. On page 257, it talks about how to approximate the maximum of a function with Laplace's method. I cannot understand how this method can be used to find the ...
Amin's user avatar
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How can I solve Laplace Tranformation of $1/s^{5/2}$?

I have just started Laplace Transformation And I came across a problem which contains $1/s^{5/2}$ How to solve it? I know $\mathcal L\{t^n\}= n!/s^{n+1}$ Please say how to solve it.
mainak mukherjee's user avatar
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1 answer
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A question on laplace transform

I try to solve the following question on Laplace transform $$L(\{ \int_0^{t}e^{-x^2}\})$$ I solved as following: $$L(\{ \int_0^{t}e^{-x^2}\})=\frac{1}{s}L(\{e^{-x^2}\})=\frac{1}{s}L(\{\sum_0^{\infty}\...
d.y's user avatar
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Asymptotics of Laplace like integral with shrinking integration intervals

I want to find the asymptotics of the integral of the form $$I(M) = \int_0^1 x^M e^{-M f(\frac{x}{\ln M})} dx$$ as $M \to +\infty$. Assume also that $f$ is analytic and increasing near $0$, with power ...
user372596's user avatar
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3 answers
274 views

Asymptotic behavior of integral with Laplace's method

I am working on the following integral $\int_0^1 dx\int_0^1 dT \sqrt{1-(1-\sqrt{x}+\sqrt{xT})^2} e^{-n xT},$ as $n\rightarrow \infty$. The goal is to find the asymptotic behavior of the integral to ...
Yu Tian's user avatar
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1 answer
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How to evaluate the following Gaussian integral by completing the summation of squares

I read this paper on parameter estimation using Bayesian theory, and I came across the following integral that I did not know how to solve. We have the following joint posterior probability for the ...
Hamad El Kahza's user avatar
2 votes
1 answer
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Laplace Transform of Frequency Decay: $\sin(2\pi t \cdot f_0 \cdot e^{-t})u(t)$

For a sine wave of exponentially decaying frequency, is the Laplace Transform numerically solvable? If not, why? One example could be: $$\sin(2\pi t \cdot f_0 \cdot e^{-t})u(t)$$ where $f_0$ is the ...
Jacob's user avatar
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1 answer
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Calculate the inverse Laplace transform by convolution

Consider the following problem: determine $\displaystyle\mathscr{L}^{-1}\bigg[\frac{s^2+1}{s^2(s^2-4s+9)}\bigg]$ using formulas and using convolutions. Using the formulas I found that the solution ...
AndVld's user avatar
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1 answer
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Show $\lim_{n \to \infty} \mathbb{E} [ f(S_n) \mid S_n \geq t] \mathbb{P}(S_n \geq t) = 0$

Let $S_n$ be a sequence of positive random variables, such that the following tail bound holds for any $n$: $$ \mathbb{P} (S_n \geq t) \leq e^{- n t} \hspace{1cm} \text{for sufficiently large $t$} $$ ...
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2 answers
178 views

Finding asymptotic growth using the Laplace Method

I am asked to find the the asymptotic growth of the integral $$I(k,m)=\int_0^1{x^{nk}}{{(1-x)}^{nm}}dx$$ with $k,m>0$ fixed. How do I approach this solution? I know it means finding the growth as $...
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Using laplace transform method (RLC) Circuit

So I been stuck in this problem for quite a while The current I(t) in a series circuit RLC is ruled by initial value: $$I''(t)+2I'(t)+2I(t)=g(t), \ \ I(0)=10; I'(0)=0$$ where $$g(t)={(20; 0< t< ...
QueroPizzadePao's user avatar
-1 votes
1 answer
54 views

Integration with Laplace transforms

I have this problem to solve, but I couldn't get the solution: $F(s) = \frac{s-2}{s^{2}-1}$ $s>1$ $\int_{0}^{+\infty}\left ( \frac{f(t)}{e^{2t}} \right )dt$ $L\left [ e^{-2t}f(t)) \right ]=F\left ( ...
Pedro R.'s user avatar
1 vote
0 answers
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Solution for 2D steady-state heat equation for rectangular plate

Considering $∇^2U=0$ for a rectangular plate with a maximum length of $a$ and width of $b$ respectively. Following are the boundary conditions: for $y=0$ $U_y = 0$ within $0<x<a/2$ and $U = -...
Unknown Name's user avatar
3 votes
1 answer
319 views

first term in the asymptotic expansion using method of steepest descent

I am working with the following intgral: $\int_{0}^{\infty}t^{n}e^{-x(t+\frac{1}{t})}dt$ as $x\rightarrow \infty$ Now, I have been trying to solve this using the method of steepest descent. After ...
L. Johnson's user avatar
1 vote
0 answers
104 views

Difference between finding asymptotics of Beta function through Laplace versus through Stirling

Im trying to find the asymptotic expression for the beta function $B(x,y) = \frac{ \Gamma(x) \Gamma(y)}{\Gamma(x+y) } $. Using Stirling's approximation $\Gamma(x) \sim \sqrt{\frac{2\pi}{x} } (\frac{x}{...
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How to you prove that the Heat Kernel of the Sub-Laplacian is given by this formula?

I been trying to show that the two integrals are equal but to no avail ,is my approach correct,I been looking at numerous literature and still could not see how those 2 integrals are equal,is one ...
scroo0ooge's user avatar
1 vote
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Laplace transform of f(t/a)

If $\mathcal{L}\{f(t)\}=F(s)$, what is the value of $\mathcal{L}\{f\left(\frac{t}{a}\right)\}$? Well, I tried the problem and got $$a \, \int_{0}^{\infty} e^{-s a u} \, f(u) \, du$$ using integration ...
sheephony priya's user avatar
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Integration of exponent of basis function

I have a problem where I need to evaluate $D=\int_a^be^{b(s)^T(c+Au)}ds $ where $b(s)$ is the B-spline. I know the Laplace method but I don't think I can use it here since it's the exponent of B-...
ash1's user avatar
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1 vote
2 answers
182 views

Asymptotic Expansion for integral with a moving maxima

$$\int_{0}^{\infty} e^{-t-x / t^{2}} d t$$ as $x \rightarrow 0$. I understand that this is a Laplace integral with a moving maximum and that I will need to rescale to get a new variable $s=tx^{-1/3}$ ...
zhizhi's user avatar
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1 answer
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Laplace Transform equation

Equation Viewing the picture attached, can anybody tell me why $$-\frac{A}{s+a}*e^{-(s+a)*t}$$ suddenly becomes $$\frac{A}{s+a}$$ in this example? What happened to our exponential equation and why is ...
Nemanja Vuksanovic's user avatar
3 votes
1 answer
162 views

Gibbs measure is concentrated in the set of global minima

So I was reading Chii-Ruey Hwang's paper called "Laplace's Method Revisited: Weak Convergence of Probability Measures" I will sort of give the basic premise of the paper: Let $Q$ be a fixed ...
rostader's user avatar
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Is there a Laplace/saddle point method extension for integrands that are null at the maximum?

I am interested in computing the asymptotics of the moments of a probability distribution defined by $p(x \mid \lambda) \propto \exp(-\lambda V(x))$ where $V$ is some smooth multivariate function of ...
Adrien Corenflos's user avatar
3 votes
0 answers
88 views

Estimate problem related to the asymptotic expension of Wallis integral

In the books Analytic combinatorics, the authors write (p. 757, eq (36)), with $k_n=n^{1/10}$, that \begin{align*} I^{(1)}_{n}:&=\int_{-k_{n}/\sqrt{n}}^{k_{n}/\sqrt{n}}\cos^n(x)\ dx\\ &=\...
Cal's user avatar
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1 vote
2 answers
582 views

Finding laplace inverse of $\frac{1}{(p+2)^2(p^2+\omega^2)}$

I am finding the laplace inverse of the below :- $$\frac{1}{(p+2)^2(p^2+\omega^2)}$$ I can apply convolution theorem by taking $$f(p)=\frac{1}{(p+2)^2}, g(p)=\frac{1} {(p^2+\omega^2)}$$ and ...
llecxe's user avatar
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3 votes
2 answers
156 views

Asymptotic expansion for the following integral

I was trying to find the asymptotic expansion for $$\int_0^1 \sqrt{t(1-t)}(t+a)^{-x} \; \mathrm{d}t,$$ for $a>0$ as $x \rightarrow \infty$. I have already tried re-writing $$(t+a)^{-x}=\exp(- x\log(...
Petals's user avatar
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1 answer
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Existance of piecewise continious function if laplace transformation is given.

How to confirm if there exists a piecewise continuous function $f(t)$ whose Laplace transform $F(s)$ is given? I know the existence theorem for the existence of Laplace transform but don't know how to ...
UJM's user avatar
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