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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Is there a simple way to evaluate the integral $\int_{\mathbb{R}^n} \delta(x_1^2+\dots+x_n^2-y)\text{d}x_1\dots\text{d}x_n$?

My approach: In general we have: $$\delta(x-x_0)=\frac{1}{2\pi}\int_{\mathbb{R}} e^{-it(x-x_0)}\text{d}t$$ Then ($y \ge 0$): $$I=\int_{\mathbb{R}^n} \delta(x_1^2+\dots+x_n^2-y)\text{d}x_1\dots\text{d}...
Leonardo's user avatar
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Non-zero expectation of $p(z)^kz$ for non-even univariate polynomial $p$ with Gaussian variable $z$.

Suppose that $p:\mathbb{R} \rightarrow \mathbb{R}$ is a polynomial and let $z$ be a standard Gaussian variable $(z \sim \mathcal{N}(0,1))$. I am interested in the behavior of the expectation $\mathbb{...
Yatin Dandi's user avatar
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Gaussian integral with complex coefficients [closed]

We are all familiar with the standard generalized Gaussian integral $$\int_{\mathbb R} dx \ \exp\left(-ax^2+bx\right)=\sqrt\frac{\pi}{a}\exp\left(\frac{b^2}{4a}\right) $$ where $a,b\in \mathbb R$. I ...
Noobgrammer's user avatar
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Relation between Gaussian integral and Kummer's hypergeometric function

Short version: I need to find out what known identity is used to convert the following integral into the formula with Kummer's confluent hypergeometric function, ${}_1F_1(a;b;z)$: $$ (2\pi\sigma^2)^{-...
dherrera's user avatar
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2 answers
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Fourier Transform of Gaussian with imaginary pole

I'm struggling at the moment with the following integral, which is essentially an inverse Fourier transform of a Gaussian with a single pole on the imaginary axis: $$I(Q) = \frac{1}{2}\int_{-\infty}^{\...
hedlejo's user avatar
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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 ...
APP's user avatar
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Propagating gaussian wavepacket

I have a problem with evaluating the following gaussian integral in two different ways \begin{equation} \psi(x,t) = \int_{\infty}^{\infty} e^{-\frac{\alpha^2 \hbar^2}{2} (k-k_0)^2} e^{(ikx - i\frac{\...
EM_1's user avatar
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Gaussian integral of squared shifted error function

The integral of a shifted error function with respect to the Gaussian measure admits a nice closed form expression: $$ \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{1}{2\sigma^{2}}(x-...
kurtachovo's user avatar
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$\left(-\frac{1}{2}\right)!$ from the Hypersphere

I was deriving the formula for volume of a n- dimensional Hypersphere when I came across something interesting. It seems like we can define $(0.5)!$ without resorting to the Gamma function. Define: $$...
Tanmay Gupta's user avatar
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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 ...
佐武五郎's user avatar
3 votes
3 answers
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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 ...
Joff's user avatar
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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 ...
Boby's user avatar
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Log-convexity of a function defined by an integral (Normal Mills ratio)

Let's define $f(x)$, for all $x>0$ by : $$f(x)=e^{x^2/2}\int_x^{+\infty}e^{-t^2/2}dt$$ I would like to prove that $f$ is log-convex, which is equivalent to the following condition : $$\forall x>...
Adren's user avatar
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Seeking Probability Function Invariant under Normal Gaussian Convolution

I'm currently working on a problem where I need to find a probability function, $P(x)$, that remains unchanged after a normal Gaussian convolution. Specifically, the function should satisfy the ...
Peyman's user avatar
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Computing Gaussian-Weighted Integrals for Small Lengthscales

I am attempting to compute the following integral: $$ I(l) = \int_{[-1,1]^2}dxdy \sqrt{1-x^2} \sqrt{1-y^2}e^{-\frac{(x-y)^2}{l^2}} $$ I am using Mathematica for symbolic calculations and translating ...
user65854's user avatar
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Generalization of Laplace's integral

A generalization of the Gaussian integral $$ \int_0^{\infty} e^{-\pi at^2} \, dt = \frac{1}{2\sqrt{a}}, \hspace{0.5cm} a>0 $$ is Laplace's integral: $$ \int_0^{\infty} e^{-\pi at^2-\pi b/t^2} \, dt ...
Dave's user avatar
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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 ...
Alireza Ghazavi's user avatar
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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}...
Izzy Tse's user avatar
4 votes
1 answer
252 views

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 $n \geq 1$ ...
Mathews Boban's user avatar
2 votes
0 answers
38 views

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. ...
matilda's user avatar
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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 ...
Alireza Ghazavi's user avatar
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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 \begin{equation} \mathbb{R}^n_+ := \{ x \in \mathbb{R}^n \mid x_i \geq 0 \text{ for all } i=1,\cdots,n\} \end{...
Keith's user avatar
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Extending Laplace transform to multivariate cases?

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 ...
Keith's user avatar
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Demonstration of an identity between a simple and a double integral of a Gaussian-Lorentzian product

In the context of a physics problem, I found that a double integral of Gaussian and Lorentzian products has values that are extremely close to that of a single integral. By defining $D(\sigma) = \frac{...
Gipfeli's user avatar
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1 answer
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Gaussian path integral over complex field

I am studying the notes of David Tong on Statistical Field Theory (https://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf). I don't understand how to formally get the result after Eq. 2.11, that is the ...
Ruth Murphy's user avatar
3 votes
0 answers
34 views

A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$

Let $\mathcal{E}(S^1)$ be the space of smooth functions on the circle $S^1$ and denote its dual as $\mathcal{E}'(S^1)$. Then, by the Minlos theorem, there exists a unique probability measure $\mu$ on $...
Keith's user avatar
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1 vote
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Manipulation into Gaussian (?) integral - Statistics

Let $\underline X$ be a sample of size $n$ from normal distribution $X \sim \mathcal{N}(\theta,\,\phi^{-1})$. First, I'm asked to show that the likelihood of the observations $x_1, x_2, ..., x_n$ is $$...
John0207's user avatar
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If $f(x) \neq 0$ on $[0,\infty)$, does there exists some $\alpha \in [0,\infty)$ such that $\int_0^\infty f(x)e^{-\alpha x^2}dx \neq 0$?

Ok,I have refined my question. Suppose that $f : [0,\infty) \to \mathbb{R}$ is measurable and $\lvert f(x) \rvert \leq x$. Further suppose that $f(x)$ is not zero almost everywhere. Then, I wonder if ...
Keith's user avatar
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Question about certain integral equality

My professor proved Wigner's semi-circle law for random matrices today in class. In part of the proof, he claimed that $$\int dA \sum_{i,j}\Big(\frac{\partial}{\partial A_{ij}}+\frac{\partial}{\...
slowspider's user avatar
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Taylor expansion for a Gaussian integral in terms of the $2$-moment?

Let $\mu$ be a centered Gaussian measure on $\mathbb{R}^N$ with the covariance matrix $\sigma$. Let $F : \mathbb{R}^N \to \mathbb{R}^N$ be some smooth, bounded mapping. Then, I wonder if it is ...
Keith's user avatar
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2 votes
0 answers
36 views

Why not use real Gaussian formulas to evaluate complex ones?

My professor has told us that it is mathematically incorrect to evaluate complex Gaussians using well-established formulas for the reals. For example, the professor says that it is incorrect to ...
FormalSymmetry's user avatar
1 vote
1 answer
95 views

gauss integral of the Phi function: $\int_{-\infty}^{\infty} \Phi(x/2+k/x) e^{-(x-a)^2/2}dx$

Is there an analytic solution for the following Gaussian integral? $$\int_{-\infty}^{\infty} \Phi(\frac{x}{2}+\frac{k}{x}) e^{-\frac{(x-a)^2}{2}}dx$$ with $a,k$ are both real numbers, $\Phi$ CDF of ...
Mingzhou Liu's user avatar
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37 views

Integral of squared exponential time elementwise polynomial

Let $z\in \mathbb{R}^D$ and $\Sigma\in\mathbb{R}^{D\times D}$ be a positive semi-definite matrix. Then I would like to compute the following integral $$ \int_{\mathbb{R}^D}\exp\left\lbrace-\dfrac{1}{2}...
cdmath's user avatar
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Approximating a modified Gaussian integral by considering tail-end behavior

I am considering calculating the following definite integral. The limits of integration are positive and $c > 0$. One can see that the integrand is bounded above by the Gaussian in the numerator ...
Debbie's user avatar
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A bit of confusion with the Gaussian measure and pushforward by a random vector

Let $d\mu$ be a centered Gaussian measure on $\mathbb{R}^n$ with the covariance matrix $\sigma$ and $X : \mathbb{R}^n \to \mathbb{R}^n$ be any measurable mapping. Or we can simply regard $X$ as a ...
Keith's user avatar
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35 views

Equivalence with the Weierstrass transform

I have the following expression $$\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(x-y) e^{-x^2/4 t} \tag{1},~~\forall ~y \in \mathbb{R}.$$. I am trying to relate it with the generalized ...
Julio Abraham Mendoza Fierro's user avatar
3 votes
1 answer
146 views

How do I prove the relation: $\intop_{t=0}^{+\infty}\frac{e^{-t}}{\sqrt{t}}dt=2\sqrt{2}\intop_{x=0}^{+\infty}{\sin(x^2)}dx$

I want to prove the following relation: $${\Gamma_{1/2}}=\intop_{t=0}^{+\infty}\frac{e^{-t}}{\sqrt{t}}dt=2\sqrt{2}\intop_{x=0}^{+\infty}{\sin(x^2)}dx$$ I noticed that: $$\frac{\intop_{x=-\infty}^{+\...
LithiumPoisoning's user avatar
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2 answers
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Limit Bessel Gaussian

I am able to prove that: $$\lim_{\varepsilon \to 0}\int^{\infty}_{0}\varepsilon^{-1}\left(J_{\frac{3}{2}}\left(\frac{r}{\varepsilon}\right)\right)^{2}\exp{(-r^{2})}r dr<\infty.$$ But I am unable to ...
LLH's user avatar
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4 votes
0 answers
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Voigt function from Convolution theorem

I am currently reading this paper on lasers by G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu. The author considered various noise sources present in the laser experiment and obtained the ...
FearlessVirgo's user avatar
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Is the kernel of the Laplacian fractional operator positive and us a Schwartz function?

Let $p_t(x):=\int_{\mathbb{R}^n} \mathrm{e}^{ix\cdot \xi}\mathrm{e}^{-t|\xi|^2}\,d\xi$ be the heat's kernel. within the properties of the kernel $p_t$, are fulfilled that $p_t(x)>0$ for all $t\geq ...
eraldcoil's user avatar
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Why is the upper bound on the gaussian integral 2pi?

If you integrate the gaussian integral using polar coordinates you can convert into the following integral: $$\int_{\theta=0}^{\theta=2\pi} \int _{r=0}^{r=\infty} e^{-r^2}r dr d\theta=I^2$$ I get the ...
Npola The Maths Guy's user avatar
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1 answer
44 views

Discrepancy wolfram alpha and hand calculation of a gaussian integral

Wolfram alpha gives me the following result: $$ \int_0^{+\infty} \frac{x^2\, e^{-a x^2}}{\sqrt{x^2 +m^2}}\, dx = \frac{\sqrt{\pi}}{4 a} U\Big(\frac{1}{2},0, a m^2 \Big)$$ and it does use the usual ...
Noix07's user avatar
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1 vote
2 answers
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Sum of Gaussian densities

I wanted to know if one could write the sum of Gaussian densities with different deviations as a Gaussian density. More specifically, For $\sigma>0$, let $f_\sigma$ be defined as :$$f_\sigma(x) \...
NancyBoy's user avatar
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3 votes
1 answer
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A specific "gaussian" integral

I wish to compute the following function $$\forall\ \mathbf{b}\in \mathbb{R}^3,\quad V(\mathbf{b}) := \int_{\mathbb{R}^{3}} \frac{ e^{-\alpha \mathbf{k}^2 + \mathbf{b}\cdot \mathbf{k} } }{ \sqrt{\...
Noix07's user avatar
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2 votes
0 answers
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Extension of single zero-crossing property

Let $f\in\mathscr{C}^2(\mathbb{R},\mathbb{R})$ a strictly increasing function, striclty convex on $(-\infty,0)$, strictly concave on $(0,\infty)$ and let $\sigma_1>\sigma_2>0$ be two real ...
NancyBoy's user avatar
  • 301
3 votes
1 answer
160 views

Zero-crossing for convolution

Suppose that you have a function $f\in\mathcal{C}^2(\mathbb{R},\mathbb{R})$. Can one show that if the function $g$ defined by: $$g(x):=\int_\mathbb{R}f(s)e^{-(s-x)^2/2}ds$$ has to zeros $x_1$ and $x_2$...
NancyBoy's user avatar
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Unique root for a simple convolution

I am struggling to show the following problem. Let $f \in \mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ (satisfying the exponential growth condition) be a real function and $x^*\in \mathbb{R}$ such that $...
NancyBoy's user avatar
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1 answer
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Gaussian integration step for exponential tail bound of Chi squared variable

I am reading a recent book on statistical learning theory and am stuck on understanding a step in the derivation of an exponential tail bound of Chi squared variables. In the following, $X_i$ is a sub-...
mtcrawshaw's user avatar
2 votes
1 answer
113 views

Sum of integrals of gaussian function

The background for this question is inferential statistics. I am trying to derive some characteristics about estimators in the context of design-based inference. Let us assume I have a population of $...
Luca Bi's user avatar
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2 votes
0 answers
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Gaussian integral and Euler's identity

I have noticed that there are two main relationships between the mathematical constants $\pi$ and $e$: The Gaussian integral, $\int_{-\infty}^{+\infty} e^{-x^2} dx = \sqrt{\pi}$ Euler's identity, $...
protarius's user avatar

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