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

766 questions
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Tighter tail bounds for subgaussian random variables

Let $X$ be a random variable on $\mathbb{R}$ satisfying $\mathbb{E}\left[e^{tX}\right] \leq e^{t^2/2}$ for all $t \in \mathbb{R}$. What is the best explicit upper bound we can give on $\mathbb{P}[X \geq x]$ for $x > 0$? A well-known upper bound…
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Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?

Is the following integral a convergent integral? Can we compute it, precisely? $$\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu $$ Here $\mu$ is the usual measure of $M_{n}(\mathbb{R})\simeq \mathbb{R}^{n^{2}}$? So $\mu$ can be counted as…
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Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a smooth function defined on $\textbf{R}^d$. What are the assumptions I should use to assume that $$\operatorname{div}\left(\nabla G(x) +xG(x)\right)=0 \quad (\forall x\in \textbf{R}^d)$$ implies that $G$ is a Gaussian? (Several answers…
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Finding the anti-derivative of $ \frac{e^{-c y^2 }}{y\sqrt{y^2-1}}$

I am trying to evaluate the integral \begin{align} \frac{1}{2\sqrt{2}\pi}\int_{0^{-}}^{t} ds \ \frac{e^{-x^2/2S^2(t,s) }}{\Sigma(s) S(t,s)} \end{align} where $S(t, s) = 2D(t-s)+\frac{\Sigma(s)}{2}$ and $\Sigma(s)= \sigma^2+2Ds$. I found a rather…
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Approximation of integral of gaussian function over a parallelepiped

Given a multi-dimensional gaussian function, defined by $$f(\boldsymbol{x})=\exp\left\{-\frac{1}{2} \boldsymbol{x}^T\boldsymbol{x} \right\}=\exp\left\{-\frac{1}{2} \sum_{i=1}^nx_i^2 \right\}$$ And an integration region as the form of a…
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Fourier transform of Hermite polynomial times a Gaussian

What is the Fourier transform of an (n-th order Hermite polynomial multiplied by a Gaussian)?
7
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Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$

I have to find $$I=\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx $$ I think we could use $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2} $$ But I don't know how. Thanks.
7
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Evaluating $ \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \mathrm{d}\tau$

I need to evaluate this integral: $$ I(t,a) = \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \ \mathrm{d}\tau $$ where the $\mathrm{erf}(\tau)$ is the error function. I can prove that this integral converges. By employing the python…
7
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2 answers

Gaussian Integral using single integration

So the Gaussian integral basically states that: $$ I = \int_{-\infty}^{\infty} e^{-x^2} \ dx =\sqrt{\pi}$$ So the way to solve this is by converting to polar co-ordinates and doing a double integration. Since I haven’t learnt double integration, I…
7
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3 answers

Gaussian type integral $\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$

When working a proof, I reached an expression similar to this: $$\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$$ I've tried the following: 1. I tried squaring and combining and converting to polar coordinates, like one…
7
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Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)

We identify $M_{n}(\mathbb{R})$ with $\mathbb{R}^{n^{2}}$ We put $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu=\lim_{r\to \infty} \int_{D_{r}} e^{-A^{2}}$ where the later is counted as a Riemann integral not Lebesgue integral. Here $D_{r}$ is the…
7
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1 answer

Gaussian Integral using contour integration with a parallelogram contour

I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function, $f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi z)}$ with a residue $1/\pi$ at $z=0$. The outline…
7
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2 answers

Higher Order Terms in Stirling's Approximation

Some websites and books give stirling approximation as $$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left(\frac{1}{n} \right)\right)$$ However when I check their derivations most do it analytically up to $$n! \approx \sqrt{2 \pi…
6
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Improvement of Jensen inequality for random variables

Jensen inequality implies that for every real random variable $X$ and every integer $n\in \mathbb N$ $$ (\mathbb E[X^2])^n \,\leq\, \mathbb E[X^{2n}]$$ by convexity of the function $x\mapsto x^n$ for $x\geq0$. Now if we apply the previous inequality…
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