# 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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### gaussian integral without the error function

here is my attempt.is it correct? $$I = \int e^{-x^2}dx$$ let $u = e^{-x^2}$ and $v=x$ then : $$I = xe^{-x^2} - \int -2x^2e^{-x^2} dx ; J = \int -x^2e^{-x^2}dx$$ here is where I'm in doubt am I alowed ...
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### Why the dummy variable $y$ in the calculation of the gaussian integral as follows?

I don't understand why you have to use a different variable when squared the first integral? It is commonly glossed over to explain this.
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### Integrating bi-variate Gaussian density with respect to their correlation coefficient

I'm wondering if there is any literature that deals with the integral of the following type $$\int \dfrac{1}{2\pi\sqrt{1-\rho^2}} \cdot \exp(-\dfrac{(a^2+b^2+2\rho ab)}{2(1-\rho^2)}) \cdot d\rho$$ ...
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### Gaussian integral in 3rd dimensions

I have been wondering about computing $(1/3)!$ and using the Gamma function. After substituting for $x=t^{\frac{1}{3}}$, I got $\int_{0}^{\infty}e^{-x^{3}}dx$. May I know if there is any way to ...
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### How to prove or calculate $E[\int_{t_{i-1}}^{t_i} e^{-μ({t_i}-s)}\sigma B_s |x_{t_{i-1}}]=0$?

$B_s$ is brownian motion. Because $\int_{t_{i-1}}^{t_i} e^{-μ({t_i} -s)}\sigma dB_s$ has a Brownian component, it is normally distributed with the mean zero according to Taylor & Karin 1998 's ...
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### Evaluating the Gaussian-like integrals $\int dx\, x^{-n} \exp(-(x-b)^2)$

Is there a known form for indefinite Gaussian integrals of the form $$\int dx\, x^{-n}\exp(-(x-b)^2)$$ where $n$ is a positive integer and $b$ is some constant? Mathematica cannot solve integrals of ...
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### Very fast but inaccurate estimations of multivariate Gaussian integral over a hypercube

$\def\Z{\mathbb{Z}}\def\R{\mathbb{R}}\def\A{\mathcal{A}}\def\N{\mathcal{N}}$ I'm working on 4D positive real values, i.e. $\R^4_{\geq 0}$, where it is gridized with hypercubes of side length $a > 0$...
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### Find flux of the vector field $\vec A$ through the surface of a sphere with radius R and center on the origin

The vector field is given by the spherical coordinates $\vec A = kr^3\,\vec {e}_r$, where $k$ is a constant and $r=\sqrt {x^2+y^2+z^2}$. I thought about using the Gauss-Ostrogradski theorem, where the ...
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### Coulomb Potential and laplace operator

How to proof the equality $\int_{S_{a}^{n-1}}\nabla u.d\sigma= c *vol (S^{n-1})$. Hi all, I am reading a text in portugues about PDE, is about Laplace operator and Coloumb Potential my specifics ...
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### How do I convert the integral $\int_0^\infty \int_0^\infty (x^2 + y^2)e^{-(x^2+y^2)} \ dxdy$ into polar coordinates?

Solve the following integral $$\int_0^\infty \int_0^\infty (x^2 + y^2)e^{-(x^2+y^2)} \ dxdy$$ For me, it is clear that we can use polar coordinates to solve this integral quickly (and yes, we can do ...
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### Gaussian integral using Euler/Jacobi theta function and $r_2(k)$ (number of representations as sum of 2 squares)

The Euler/Jacobi theta function (using the notation of this question) is $\vartheta_3(\tau) := \sum_{n\in \mathbb Z} q^{n^2}$ where $q = e^{2\pi i\tau}$ is the nome. The square $(\vartheta_3(\tau))^2$ ...
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### Does the expectation of any bounded function on Gaussian variables exist?

I want to prove the expectation of the function $f(X)$ exists, where $X$ is a Gaussian variable. Given that $f(X)$ is bounded between $[-c, c]$, with $c$ being some positive constant, does the ...
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### Computing the Fourier Transform of $\exp(-\log(x)^2)$

I would like to use the approach of R. E. Fredericksen and R. F. Hess, “Estimating multiple temporal mechanisms in human vision,” Vision Res., vol. 38, no. 7, pp. 1023–1040, Apr. 1998, doi:10.1016/...
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I'm trying to solve the following integral: $$\int_{x} \int_{y} \int_{z} \int_{w} \exp \left(-\frac{1}{2}\left\|\left(\begin{array}{l} y-x \\ z-y \\ y-z \end{array}\right)\right\|^{2}\right) d x d y ... • 187 0 votes 0 answers 38 views ### Find integration limit given the value of the (2D) integral (Previously asked on Stackoverflow - now deleted - it was pointed out that it makes more sense to ask my question here.) I am trying to solve for R_j(I_0) given in Eq. 12 of this paper (see also Eq. ... • 323 0 votes 1 answer 16 views ### What is the second step of this standard normal MGF derivation? On this page, it shows the derivation for the MGF of a standard normal density. I don't get why the second step has a \sqrt{2}\sigma which is pulled out from the integral. Can anyone help me ... • 603 0 votes 2 answers 95 views ### Numerical integration of \exp(-x^2) in a bounded region Is there a possible way to integrate \exp(-x^2) in a bounded 2D triangular region numerically with minimal number of Gauss points? The Gauss-Hermite quadrature scheme is suitable for unbounded ... -1 votes 1 answer 34 views ### About gaussian distribution - calculation of variance (change of variables step) According to these slides, in slide 3: Why we do not change the x of exp with (x-mu)^2 ? I did not understand this part. • 109 0 votes 1 answer 46 views ### Solving Gaussian Integral I learned that a Gaussian Integral is $$\int_{-\infty}^{\infty}xe^{-2ax^2}dx=0$$ Because of the odd function symmetry. But if I shift the x to \int_{-\... 1 vote 1 answer 53 views ### Limit of integral with Parseval I am trying to evaluate the limit$$ \lim_{n \to \infty} \int_{-\infty}^{\infty} \frac{\sin(nt)}{\pi t}f(t) \,dt, $$where$$ f(t)=e^{-t^2+2t}. $$Since I am working with Fourier transforms I thought ... • 133 3 votes 0 answers 106 views ### Is the following cartoon mathematically correct? I saw the following cartoon about probability and statistics the other day: The joke in the above picture being that "the top 1%" (i.e. the integral of the gaussian probability distribution ... • 1,688 0 votes 0 answers 66 views ### How do you integrate the volume of 3 two-dimensional gaussian distributions with 4 hyperbole boundaries rotated 90 degrees at various intervals? Suppose you have 3 two-dimensional gaussian distributions added up on each other pushed to be within 4 identical hyperbolas rotated in relation to each other by 90 degrees and their most proximal ... 1 vote 1 answer 44 views ### When evaluating a definite integral by integrating by parts is the following allowed?$$= \int_a^b f(x)g(x) \, dx = \int_a^b f(x)\, dx\cdot [g(x)]_a^b\, -\int_a^b \bigg[ g'(x) (\int f(x)dx\bigg] \, dx  = C - \int_a^b \Bigg[g'(x)\cdot\int_a^b f(x)dx\, \Bigg]dx \, $$where C is ... • 11 0 votes 1 answer 38 views ### Derive the quadrature rule of the form \int_{-1}^1 f(x) dx \approx w_0 f(\alpha) + w_1 f'(\beta) From a practice problem Problem Statement Let the interval [a,b] be divided into n subintervals by n+1 points x_0 = a , x_1 = a+h, x_2 = a+2h , \dots ,x_n = a+nh =b. Derive the quadrature ... • 744 1 vote 0 answers 9 views ### Relationship between \gamma_n(A + B), \gamma_k(A), and \gamma_{n-k}(B), where A \subseteq V, B \subseteq V^\perp, and \gamma_n := N(0,I_n) Let \gamma_n = \gamma_1^{\otimes n} be the standard gaussian measure on \mathbb R^n and let V be a k-dimensional subspace of \mathbb R^n with orthogonal completment V^\perp. Let A and B... • 8,309 0 votes 0 answers 23 views ### Integrating the gaussian kernel on unbounded curves in \Bbb C It is well known (e.g. using Fourier transform) that if \beta\in\Bbb C with \Re\beta>0, then$$ \int_{\Bbb R}e^{-bu^2}\,du=\sqrt{\frac{\pi}{\beta}}\;. $$Once chosen a branch for the square ... • 11.4k 0 votes 0 answers 24 views ### Given |\int f(\zeta)\,d\zeta-1|<\epsilon, what can we say about \int|f(\zeta)|\,d|\zeta|? Consider the following unbounded curve in \Bbb C$$ \Gamma=\{x+ih(x),\;\;x\in\Bbb R\} $$where h\colon\Bbb R\to\Bbb R is a bounded \mathscr C^1 real function, such that h'(x)\to0 as |x|\to\... • 11.4k 1 vote 0 answers 70 views ### How do I compute this integral on the set A? I have the following problem: Compute the integral \int_{\partial A} \langle F,\nu \rangle dS where \nu is the normal vector and$$F(x,y,z)=(2x-3y+z,\,\,\,x-y-z,\,\,\,-x+y+2z)$$and$$A=\{(x,y,z):...
After thinking and searching of/for an answer for an eternity, you are my last hope: The given equation: $$rot(µ^{-1}rot(\vec A))-\epsilon\omega^2*\vec A=\vec J$$ should be multiplied with a ...