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.

The Gaussian integral or, Euler–Poisson integra or, the probability integral , closely related to the erf function, appears in many situations in engineering mathematics and statistics. It can be defined by

$$I(\alpha)=\int_{-\infty}^{+\infty}e^{-\alpha ~x^2}~ dx$$

Laplace $~(1778)~$ proved that $$\int_{-\infty}^{+\infty}e^{- ~x^2}~ dx=\sqrt{\pi}$$

Applications:

The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution. The same integral with finite limits is closely related to both the error function and the cumulative distribution function of the normal distribution. In physics this type of integral appears frequently, for example, in quantum mechanics, to find the probability density of the ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator, and in statistical mechanics, to find its partition function.

Some other forms:

$$\int_{-\infty}^{+\infty}e^{-\alpha ~x^2}~ dx=\sqrt{\frac{\pi}{\alpha}}$$

$$\int_{0}^{\infty}e^{-\alpha ~x^2}~ dx=\frac{1}{2}~\sqrt{\frac{\pi}{\alpha}}$$

$$\int_{-\infty}^{+\infty}e^{-\alpha ~x^2+bx}~ dx=e^{\frac{b^2}{4\alpha}}\sqrt{\frac{\pi}{\alpha}}$$

$$\int_{0}^{+\infty}e^{i\alpha ~x^2}~ dx=\frac{1}{2}~\sqrt{\frac{i~\pi}{\alpha}}$$

$$\int_{0}^{+\infty}e^{-i\alpha ~x^2}~ dx=\frac{1}{2}~\sqrt{\frac{\pi}{i~\alpha}}$$

References:

https://en.wikipedia.org/wiki/Gaussian_integral

http://mathworld.wolfram.com/GaussianIntegral.html

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A certain Gaussian integral

Can somebody evaluate the following Gaussian integral? $$ I(t,\sigma) := \int_{-\infty}^\infty \frac{dx e^{-x^2/(2\sigma^2)}}{\sqrt{2\pi \sigma^2}} \frac{\sin{\left(2 t\sqrt{1+x^2} \right )}}{\sqrt{1+x^2}} \tag{1} $$ Where of course $\sigma>0$ (and…
lcv
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Integral over the hypersphere

Assume I have a diagonal matrix $L$ of size $n$. I want to compute the following integral: $$I_n(L) \equiv \int_{(\mathbb{S}^{n-1})^2} \mathrm{d}\sigma(x) \mathrm{d}\sigma(x') \exp[n x^\top L \, x']$$ In which $\mathbb{S}^{n-1}$ is the unit sphere…
seamp
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Integral involving erf and exponential

Problem I would like to compute the integral: \begin{align} \int_{0}^{+\infty} \text{erf}(ax+b) \exp(-(cx+d)^2) dx \tag{1} \end{align} I have been looking at this popular table of integral of the error functions, and also found here the following…
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How do we evaluate the closed form for $\int_{-\infty}^{+\infty}{(-1)^{n+1}x^{2n}+2n+1\over (1+x^2)^2}\cdot e^{-x^2}\mathrm dx?$

Proposed: $$\int_{-\infty}^{+\infty}{(-1)^{n+1}x^{2n}+2n+1\over (1+x^2)^2}\cdot e^{-x^2}\mathrm dx={\sqrt{\pi}\over 2^{n-2}}\cdot F(n)\tag1$$ Where is n integer, $n\ge1$ I am struggled to find the closed form for $(1)$ Where $F(1)=1, F(2)=3,…
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The integral $\int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^x)^2} dx$.

Let $$T(n) = \int_{-\infty}^\infty \frac{e^{-x^2}}{(1+e^{x})^n} dx.$$ We have that $$ T(0) = \sqrt{\pi} \text{ and } T(1) = \frac{\sqrt{\pi}}{2}$$ and also that $$ T(3) = \tfrac{3}{2} T(2) - \frac{\sqrt{\pi}}{4}.$$ Can we find a closed form for…
6
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Computing the integral $\int \exp(ix^2) dx$

I'm trying to compute the following integral: $I_1 = \int_{-\infty}^\infty \exp(iu^2) du$. This is what I did, but it is wrong and I don't know why: $$I_1^2 = \left (\int_{-\infty}^\infty \exp(iu^2) du \right)^2 = \int_{-\infty}^{\infty}…
Tom
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How do I symbolically compute $\int_{a}^{b} e^{x^2}(\textrm{erf}(x) - \textrm{erf(a)})\;\textrm{d}x$?

I want to symbolically write (in the form of a series), the integral of: $$ \int_{a}^{b} e^{x^2}(\textrm{erf}(x) - \textrm{erf(a)})\;\textrm{d}x, \text{where }\{x, a, b\} \subset \mathbb{R} $$ The $\textrm{erf}$ function is: $$…
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Solve: $\int_{0}^{\infty}{ \left( \cos \left( ... \right)+\sin \left( ... \right) \right) \frac{e^{-u^2}}{(u^2+ \frac{\alpha}{4t})^2} }~\mathrm{d}u$

I have a really nasty integral to solve as follow: (It is good for challenge lovers!) $$ I(t)=\int_{0}^{t}{{\bigg(\cos\left({\frac{\gamma}{4(t-r)}}\right)+\sin\left({\frac{\gamma}{4(t-r)}}\right)\bigg)} e^{- \frac{\alpha}{4r}}…
5
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One Dimensional Gaussian Integral involving a rational function

In the process of solving this question Integral involving product of arctangent and Gaussian, I've come across the integral: $$ \int_0^b \frac{e^{-s^2}}{a^2 + s^2} d s . $$ This integral appears simple enough that I would expect something to be…
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Polar Coordinate Transformation - Motivation

I am trying to work out the reason why the integral $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{e^{-(x^2+y^2)}}\,dx\,dy $$ is, in polar coordinates, $$ \int_{-\infty}^{\infty}{e^{-r^2}} \,r\,dr\,d\theta$$ As I understand it, a polar coordinate…
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Covariance of a rectified (relu) Gaussian

Given a normal random vector $$X\sim N(\mu,\Sigma)$$ for spd $\Sigma$, I'm interested in the covariance matrix $K=\mathrm{cov}(Y)$ of the variable $$Y = \mathrm{relu}(X)$$ where the relu is performed elementwise $Y_i = \mathrm{Max}(0,x_i)$, so $Y$…
5
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Representing the determinant of a Hermitian matrix as an integral

Let $M=\left (\omega\mathbb{I}-A\right )\left(\omega^{*}\mathbb{I}-A^{\dagger}\right)$ be a Hermitian matrix of size $n\times n$ where $A$ is a real non symmetric matrix and $\omega=a+\mathrm{i}b$. $A^{\dagger}$ represents the conjugate transpose…
5
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What is the purpose of $\frac{1}{\sigma \sqrt{2 \pi}}$ in $\frac{1}{\sigma \sqrt{2 \pi}}e^{\frac{(-(x - \mu ))^2}{2\sigma ^2}}$?

I have been studying the probability density function... $$\frac{1}{\sigma \sqrt{2 \pi}}e^{\frac{(-(x - \mu ))^2}{2\sigma ^2}}$$ For now I remove the constant, and using the following proof, I prove…
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How to evaluate the integral $\int\mathbf{g}^T\mathbf{v} \exp{(-\frac{1}{2}\mathbf{v}^T \mathbf{A}\mathbf{v})}\,d\mathbf{v}$

I have the following multidimensional Gaussian integral: $$\int\textbf{g}^T\textbf{v} \exp{\left(-\frac{1}{2}\textbf{v}^T \textbf{A}\textbf{v}\right)}\,d\textbf{v},$$ where $\textbf{A}$ is a real symmetric $W\times W$ matrix, $\textbf{v} \in…
jjepsuomi
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Gaussian-trigonometric definite integral $\int_0^\infty \frac{e^{-x^2}}{1+a \cos x}dx$

Is it possible to evaluate this integral in closed form? $$I(a)=\int_0^\infty \frac{e^{-x^2}}{1+a \cos x}dx$$ $$0
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