Questions tagged [gaussian-integral]

For questions regarding the theory and evaluation of the Gaussian integral, also known as the Euler–Poisson integral is the integral of the Gaussian function $~e^{−x^2}~$ over the entire real line. . It is named after the German mathematician Carl Friedrich Gauss.

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Multi-mode Gaussian integral with a symmetric complex matrix in the exponent

I am trying to evaluate a multi-dimensional Gaussian integral of the form, $$\int_{\mathbb{R}^{n}} d^{n}x e^{-\frac{1}{2}x^{T} A x},$$ where the matrix $A$ is, Complex ...
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Laplace's Approximation for Integral from 0 to Infinity

Background I am working on a problem where I need to evaluate an integral from 0 to infinity and compare the numerical solution to an analytical approximation using Laplace's method. However, I am ...
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How to show that the Gaussian sequence satisfies the sifting property of the delta function?

I am trying to show that the Gaussian sequence of functions, defined by $g_a (x) = (\frac{1}{a\sqrt{\pi}}) e^\frac{-(x-x_0)^2}{a^2}$ satisfies the sifting property of the $\delta$ - function, namely ...
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Indefinite Integral Of A Wave Packet

So, I have the following function (also known as a wave packet in physics): $$5{e}^{-\frac{{x}^{2}}{25}} \cos(3x)$$ Now, I want to find the indefinite integral of this and I started with integration ...
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Integral involving erf

I would greatly appreciate if I could get some help in the following integral: $$\int_{0}^{\infty}\frac{\operatorname{erf}\left(x\right)}{a^{2} +x^{2}}{\rm d}x$$ A similar post was this one. The ...
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Extension of Dirichlet Divisors in Gauss Circle Integral Point Problem

Dirichlet Divisors A well-known problem in number theory is to study the sum of divisor functions $d(n)$: $D_2(x)=\sum_{n\leq x} d(n)$, so $D_2(x)$ can be expressed as the number of the integer points ...
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How to evaluate these Gaussian integrals?

I am reading this paper by Gross and Mezard. In it they calculate the number of solutions to the Thouless-Anderson-Palmer mean field equations for a spin glass model. As an intermediate step (...
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Using Jacobi-Anger expansion to approximate a Gaussian integral

I am trying to perform the following integral, $$\mathcal{I}=\int_{-\infty}^\infty e^{-x^2-ib y \sinh{\left(\frac{x}{b}\right)}}dx$$ with $b$ and $y$ real and $b> 0$. ...
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Integral of the product of a Gaussian and a exponential of a hyperbolic function

In a derivation I am working on, I have encountered an integral of the form $$\int_{0}^{\infty}e^{-a x^2-b\ \textrm{cosh}(x)}\ dx$$ with $a$ and $b$ real and positive. I am ...
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Contourn integration with gaussian-like argument

I'm currently working on evaluating the following integral: $$I=\int_{-\infty}^{+\infty}\frac{a-x^2+b}{(a-x^2)^2}e^{-i c x^2}dx,$$ where $a,b,c$ are real and positive constants. I wish to compute ...
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Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x\,y\,\exp\left({-\frac{1}{2}\left(x-y\right)^2-\frac{y^2}{2}}\right)\,\mathrm{d}x\mathrm{d}y$

I'm currently practicing some problems in probability theory to give me a good base for stochastic processes. Here, probability density for $Y$ is a Gaussian with zero mean and that for $X$ is a ...
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Does $\int_{-\infty}^\infty \int_0^x e^{y^2/2} \, dy \; e^{-x^2/2} \, dx$ exist?

I am trying to figure out if the following integral exists $$\int_{-\infty}^\infty \int_0^x e^{y^2/2} \, dy \; e^{-x^2/2} \, dx$$ For higher potentials in the exponent of the exponential, say $x^4$, ...
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Showing a bivariate Gaussian vector has normal components by computing the marginals

Let $( X_1 , X_2 )$ be a bivariate normal random variable. Assume that the correlation coefficient $|\rho[ X_1 , X_2 ] | \ne 1$. Show that $X_1$ and $X_2$ are Normal random variables by calculating ...
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$\int_{\mathbb{R}^2} e^{-(x^2+y^2)}=[\int_{\mathbb{R}} e^{-x^2}]^2,$ provided the first of these integrals exists. Munkres Analysis on Manifolds

I am reading "Analysis on Manifolds" by James R. Munkres. (a) Show that $$\int_{\mathbb{R}^2} e^{-(x^2+y^2)}=[\int_{\mathbb{R}} e^{-x^2}]^2,$$ provided the first of these integrals exists. ...
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$\int x^2\exp(-x^2/2\sigma^2)\rho(x)dx\leq C\sigma^2 \int\exp(-x^2/2\sigma^2)\rho(x)dx$

For a 'reasonable' pdf, $\rho(x)$ ($\mathbb{P}(A)=\int_A\rho(x)dx$), I am trying to prove the above inequality. In general, this inequality isn't true since we may take $\rho$ to have support only ...
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Propagating gaussian wavepacket

I have a problem with evaluating the following gaussian integral in two different ways \psi(x,t) = \int_{\infty}^{\infty} e^{-\frac{\alpha^2 \hbar^2}{2} (k-k_0)^2} e^{(ikx - i\frac{\...
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What does $\int_{B_r} h$ mean? Problem 3-41 in "Calculus on Manifolds" by Michael Spivak. $\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$.

I am reading "Calculus on Manifolds" by Michael Spivak. I solved (a) and (b). Then, I proved $\int_C h=\int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} rg(r,\theta)d\theta dr$ holds. My question ...
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Integrating a product of Gaussian kernels

If we consider the Gaussian kernel $k(x, y) = \exp(-\frac{\Vert x - y \Vert^2}{2\sigma^2})$, then is it true that... $$\int k(x_i, x)k(x_j, x)dx = k(x_i, x_j)$$ It seems I have seen this before, but ...
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Given a multilinear polynomial $g$, and an odd function $f$ show that $\int f(x_i) \exp(-\| x \|^2) g(x-y) d x =0, \forall y,i$ where $g(x)=c$)

Suppose that the following convolution equation holds: for all $y\in \mathbb{R}^n$ and every $i =1 ..n$ $$\int f(x_i) \exp \left(- \frac{\| x \|^2}{2} \right) g(x-y) d x =0$$ where $f(x)$ is an ...
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What is the integral $\int_{-\infty}^{+\infty} x^{-2} e^{-x^2}\,dx$?

In my research, I reached to the $$\int_{-\infty}^{+\infty}\!\! \frac{1}{\sqrt{2 \pi}x^2}e^{-x^2/2}\, dx.$$ I have tried polar change of systems, integral by parts and change of variables but I did ...
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A partition function problem under conserved total momentum, involving Gaussian integration under a delta constraint

Recently I learned about the following expression showing the kinetic part of the partition function in an N-atom ideal gas under conserved total momentum: \begin{aligned} Q_{\mathrm{Kin}}^{\mathrm{CM}...
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Integral over a small set is small and dimension independent

Let $\mu_n$ be the standard normal distribution over $\mathbb{R}^n$. Is the following true? For all $\epsilon > 0$ and $d \in \mathbb{N}$, there exists $\eta > 0$ such that for all $n \geq 1$ ...
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Is the function $\int_0^{1} e^{-t H}e^{-2x^2}dt$ in $L^2$ with respect to the gaussian measure?

Consider the function $g(x)=\int_0^{1} e^{-t H}e^{-2x^2}dt$, where $H$ is the differential operator $x\partial_x-\partial_x^2$. Is $g(x)$ in the $L^2$ space with respect to the gaussian measure; i.e. ...
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a special type of Gaussian Integral (reciperocal Quadratic functions with Gaussian Integral)

I have tried to calculate the following integral in terms of $\sigma^2$ : $I=\int_{x=-\infty}^{+\infty}\frac{1}{\sqrt{1+x^2}}e^{-x^2/{2\sigma^2}} dx$ However by using conventional methods such as ...
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If $f \neq 0$, do we have $\int_{ x_i \geq 0} f(x) e^{-\sum_{i=1}^n c_i x_i^2} d^nx \neq 0$ for some $c \in (0,\infty)^n$?

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a bounded measurable mapping. Next, define \mathbb{R}^n_+ := \{ x \in \mathbb{R}^n \mid x_i \geq 0 \text{ for all } i=1,\cdots,n\} \end{...
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Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a measurable $L^\infty$ function. Assume further that $f$ is not zero a.e. and $f(-x)=f(x)$ for a.e. $x \in \mathbb{R}^n$. Then, I wonder if we can conclude ...