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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Divergence theorem how do you calculate the volume integral, flow?

Hello I really need a thorough answer to one task which is the following one: Consider the vector field $$F(x, y, z) = (xy^2 z, x^2yz, −z),$$ the cylindrical-shape Area $$V = \left\{(x, y, z): \sqrt{x^2+y^2} < 5 \text{ and } 0 < z < 4\right\}…
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Is this definition of a Gaussian distribution correct?

Is it correct to say for the following: $$u_{k}\sim N(0,\gamma ^{2})$$ that "$u_{k}$ follows the Gaussian distribution with a mean of zero and a covariance of $\gamma ^{2}$" ? I am a bit sceptical regarding the term covariance. Shouldn't that be…
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Can't solve double integral with Gauss mark

evaluate the integral $$\int \int_D [x+y]dA \qquad , D=[1,3]\times[2,5]$$ (let [x] denote the greatest integer in x )
ekfqlc
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Infinite sum of gaussian exponential

Does anybody know a closed expression for: $$\sum_{n=-\infty}^{+\infty} e^{-(a+bn)^2}$$
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Trouble proving Gaussian-like integral

I came across Gaussian integrals, and was trying to prove them myself. I proved the basics, but am stuck on the following $$\int\limits_{-\infty}^{\infty}xe^{-a(x-b)^2}dx = b \sqrt{\frac{\pi}{a}}$$ I am having trouble thinking of how to attack this…
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integral -e^2 with variables in limits

I've been looking and maybe not fully understanding gaussian integral but what happens when there are variables in the upper/lower bounds(limits)? for example: $$\int_{x^2}^{-x^2} e^{-(x-2t)^2} dt$$ does gaussian integral there still apply here?…
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How to evaluate this integral from -infinity to infinity (improper integral)?

I'm having difficulty evaluating this integral: $\int_{-\infty}^{\infty} {x^6 e^{-x^2}} dx$ All I've been able to do is separate them and evaluate them separately although I haven't been able to successfully do that: $\int_{-\infty}^{0} {x^6…
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How can the following Gaussian integral be calculated?

How can we compute this integral? $$\int_{1.96}^{\infty} e^{\frac{-x^2}{2}} \ dx$$
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Why $\int_{- \infty} ^\infty x e^{-ax^2}dx=0$

I really don't understand why $\int_{- \infty} ^\infty x e^{-ax^2}dx=0$. I can see $x e^{-ax^2}$ as $\frac{d}{dx}\frac{e^{-ax^2}}{-2a}$ that, integrated, is clearly different from 0...
sunrise
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Empirical Process estimation using gaussian density and specific random generator

EDITED: To formulate into math framework: I have a sampling generator producing IID gaussian. To highlight the convergence in the distribution, I calculate the following error. Given a precision step m and a bandwitdh H, a theoritical density D, The…
quantCode
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Evaluation on a basis of gaussian integral

Knowing that $$\int_{-\infty}^\infty e^{-x^2} dx= \pi^{\frac{1}{2}}$$ Find: $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{\frac{-x^2}{2}} dx$$ And my question is how does this help if have the value of gaussian integral?
mkropkowski
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CDF of Gaussian Distribution Raised to a Power

Say X ~ $\mathcal{N}(0,1)$ what would be the CDF of Y=X^a (where a is an integer). I have seen how to get the PDF which would theoretically give the CDF but I'm not sure how to get a closed form solution, say using erf() like the standard normal cdf…
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what is the closed form for the inverse Laplace transform of Gaussian function $\mathcal{L^{-1}}\left\{e^{-x^2}\right\}$?

it's easy to get the laplace transform of Gaussian function as the following: $\mathcal{L}\left\{e^{-x^2}\right\}$ $=\int_0^\infty e^{-x^2-sx}~dx$ $=\int_0^\infty e^{-(x^2+sx)}~dx$ $=e^\frac{s^2}{4}\int_0^\infty…
zeraoulia rafik
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how to solve Gaussian integrals with three easy cases

I am using the the book called street mathematics to learn more about dimensional analysis. I am trying to understand a problem in the book. The question is to use dimensional analysis to find the solutions for Gaussian integral. I tried to…
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