# 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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### 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 ...
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• 181
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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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### 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 ...
• 531
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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 ...
• 531
1 vote
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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 ...
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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, ...
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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 ...
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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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• 11
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• 681
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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 ...
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 ...
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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-...
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### 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 $s=tx^{-1/3}$ ...
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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 ...
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### 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 ...
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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 ...
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### 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\\ &=\...
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### 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 ...
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### 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(...
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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 ...