Gaussian integral Definition and 65 Threads

I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?

I am looking for the solution of a multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are p $\times$ 1 and T is a positive definite symmetric p $\times$ p matrix. The integral is the...

Hi, I have recently learned the technique of integration using differentiation under the integral sign, which Feynman mentioned in his “Surely You’re Joking, Mr. Feynman”. So, I decided to try it on the Gaussian Integral (I do know the standard method of computing it by squaring it and changing...

hi guys i am trying to solve the Gaussian integral using the power series , and i am suck at some point : the idea was to use the following series : $$\lim_{x→∞}\sum_{n=0}^∞ \frac{(-1)^{n}}{2n+1}\;x^{2n+1} = \frac{\pi}{2}$$ to evaluate the Gaussian integral as its series some how slimier ...

Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average: $$ I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)} $$ for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...

I'm trying to solve the inequality: $$ \int \limits_0^1 e^{-x^2} \leq \int \limits_1^2 e^{x^2} dx $$I know that $\int \limits_0^1 e^{-x^2} \leq 1$, but don't see how to take it from there. Any ideas?

Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do...

Hello! Starting from a gaussian waveform propagating in a dispersive medium, is it possible to obtain an expression for the waveform at a generic time t, when the dispersion is not negligible? I know that a generic gaussian pulse (considered as an envelope of a carrier at frequency k_c) can be...

I am trying to evaluate the following integral. ##\displaystyle{\int_{-\infty}^{\infty}f(x,y)\ \exp(-(x^{2}+y^{2})/2\alpha)}\ dx\ dy=1## How do you do the integral above?

Consider the partition function ##Z[J]## of the Klein-Gordon theory ##Z[J] =\int \mathcal{D}\phi\ e^{i\int d^{4}x\ [\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}+J\phi]} =\int \mathcal{D}\phi\ e^{-i\int d^{4}x\ [\frac{1}{2}\phi(\partial^{2}+m^{2})\phi]}\ e^{i\int d^{4}x\...

Hey, folks. I'm doing a problem wherein I have to evaluate a slight variation of the Gaussian integral for the first time, but I'm not totally sure how to go about it; this is part of an integration by parts problem where the dv is similar to a gaussian integral...

Homework Statement I'm encountering these integrals a lot lately, and I can solve them because I know the "trick" but I'd like to know actually how the cartesian to polar conversion works: ##\int_{-\infty}^{\infty}e^{-x^2}dx## Homework Equations ##\int_{-\infty}^{\infty} e^{-x^2} = I##...

Hi everyone, in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like: I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)} where a,b,c are...

Hi all, so I've come across the following Gaussian integral in QFT...but it has a denominator and I am completely stuck! \int_{0}^{\infty} \frac{dx}{(x+i \epsilon)^{a}}e^{-B(x-A)^{2}} where a is a power I need to leave arbitrary for now, but hope to take between 0 and 1, and \epsilon is...

Homework Statement I have to prove that I(a,b)=\int_{-\infty}^{+\infty} exp(-ax^2+bx)dx=\sqrt{\frac{\pi}{a}}exp(b^2/4a) where a,b\in\mathbb{C}. I have already shown that I(a,0)=\sqrt{\frac{\pi}{a}}. Now I am supposed to find a relation between I(a,0) and \int_{-\infty}^{+\infty}...

So I've seen this type of integral solved. Specifically, if we have ∫e-i(Ax2 + Bx)dx then apparently you can perform this integral in the same way you would a gaussian integral, completing the square etc. I noticed on wikipedia it says doing this is valid when "A" has a positive imaginary part...

Homework Statement Let a,b be real with a > 0. Compute the integral I = \int_{-\infty}^{\infty} e^{-ax^2 + ibx}\,dx. Homework Equations Equation (1): \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi} Equation (2): -ax^2 + ibx = -a\Big(x - \frac{ib}{2a}\Big)^2 - \frac{b^2}{4a}The Attempt...

Homework Statement I need to evaluate the following integral: \sqrt{\frac{2}{\pi}}\frac{\sigma}{\hbar}\int\limits_{-\infty}^{\infty}p^2 e^{-32\sigma^2(p-p_0)^2/\hbar^2}\,dp Homework Equations Integrals of the form...

Homework Statement I'm trying to solve the Gaussian integral: \int_{-∞}^{∞}xe^{-λ(x-a)^2}dx and \int_{-∞}^{∞}x^2e^{-λ(x-a)^2}dxHomework Equations I can't find anything online that gives the Gaussian integral of x times the exponential of -λ(x+(some constant))squared. I was hoping someone here...

Homework Statement I am asked to evaluate ##\displaystyle\int_{-\infty}^{\infty} 3e^{-8x^2}dx## Homework Equations I know ##\displaystyle\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}## The Attempt at a Solution based on an example in the book it seems a change of variables...

I'm reading a book on Path Integral and found this formula \int_{-\infty}^{\infty }e^{-ax^2+bx}dx=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}} I Know this formula to be correct for a and b real numbers, however, the author applies this formula for a and b pure imaginary and I do not understand why...

Is there a formula for this gaussian integral: $$int_{-\infty}^{\infty}{x^4}{e^{-a(x-b)^2}}dx$$ I've tried wikipedia and they only have formulas for the integrand with only x*e^... not x^4e^... Wolframalpha won't do it either, because I actually have an integral that looks just like that...

Is there a formula for Gaussian integrals of the form $$\int_{-\infty}^{\infty}{x^n}{e^{-a(x-b)^2}}dx$$ I've looked all over, and all I could find were formulas saying $$\int_{-\infty}^{\infty}{e^{-a(x-b)^2}}dx=\sqrt{\frac{\pi}{a}}$$ and...

Hey, I am rather stuck on this gaussian integral... I have come this far, and not sure what to do now: [tex]\int dh_{01}(\frac{h_{01}}{\sigma})^{2}+\frac{\Delta k^{2}(t-x)^{2}h_{01}}{2}-ik_{0}(t-x)h_{01}[\tex] [tex]\int...

Homework Statement Show in detail that: \sigma_{x}^{2} = \int_{-\infty}^{\infty} (x -\bar{x})^{2} \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-X)^{2}}{2\sigma^{2}}} = \sigma^{2} where, G_{X,\sigma}(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-X)^{2}}{2\sigma^{2}}} Homework Equations \int u...

Homework Statement The integral of (x^n)(e^(-a*x^2)) is easier to evaluate when n is odd. a) Evaluate ∫(x*e^(-a*x^2)*dx) (No computation allowed!) b) Evaluate the indefinite integral of x*e^(-a*x^2), using a simple substitution. c) Evaluate ∫(x*e^(-a*x^2)*dx) [from o to +inf] d)...

I'm dealing with multivariate normal distributions, and I've run up against an integral I really don't know how to do. Given a random vector \vec x, and a covariance matrix \Sigma, how would you go about evaluating an expectation value of the form G=\int d^{n} x \left(\prod_{i=1}^{n}...

Homework Statement We define I_{n} = \int_{-∞}^{∞}x^{2n}e^{-bx^{2}}dx, where n is a positive integer. Use integration by parts to derive:I_{n}=\frac{2n-1}{2b}I_{n-1} Homework Equations Parts formula. The Attempt at a Solution So I'm just stuck here, I'm baffled and confused. Firstly if I...

I am trying to calculate the functional for real scalar field: W[J] = \int \mathcal{D} \phi \: exp \left[{ \int \frac{d^4 p}{(2 \pi)^4}[ \frac{1}{2} \tilde{\phi}(-p) i (p^2 - m^2 +i \epsilon) \tilde{\phi}(p)} +\tilde{J}(-p) \tilde{\phi}(p)] \right] Using this gaussian formula...

Homework Statement I'm reading Hinch's perturbation theory book, and there's a statement in the derivation: ...\int_z^{\infty}\dfrac{d e^{-t^2}}{t^9}<\dfrac{1}{z^9}\int_z^{\infty}d e^{-t^2}... Why is that true?Homework Equations The Attempt at a Solution Homework Statement Homework Equations...

If I had an integral \int_{-1}^{1}e^{x}dx Then performing the substitution x=\frac{1}{t} would give me \int_{-1}^{1}-e^\frac{1}{t}t^{-2}dt Which can't be right because the number in the integral is always negative. Is this substitution not correct? Sorry if I am being very thick but I...

I need to work out an expression for the average of a Dirac delta-function \delta(y-y_n) over two normally distributed variables: z_m^{(n)}, v_m^{(n)} So I take the Fourier integral representation of the delta function: \delta(y-y_n)=\int \frac{d\omega}{2\pi} e^{i\omega(y-y_n)} =\int...

We all know about the famous equation: \int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}. How about \int_{-\infty}^\infty e^{-x^4} dx? Or, in general, can we calculate any integral in the form \int_{-\infty}^\infty e^{-x^n} dx, where n is an even counting number?

Here is a link to a course which i am studying, http://quantummechanics.ucsd.edu/ph130a/130_notes/node89.html#derive:timegauss My problem comes from the k' term attached to Vsub(g) (group velocity). I used the substitution k' = k - k(0), factored out all exponentials with no k'...

\int_0^\infty e^{-x^2}dx \int_0^\infty e^{-y^2}dy = \int_0^\infty \int_0^\infty e^{-(x^2+y^2)} dxdy Under what conditions we could do the same for other functions? I don't get how Poisson (or Euler, or Gauss, whoever that did this for the first time) realized that this is true. It looks...

The following problem arises in the context of a paper on population genetics (Kimura 1962, p. 717). I have posted it here because its solution should demand only straightforward applications of tools from analysis and algebra. However, I cannot figure it out. Homework Statement Let z = 4...

I'm trying my very best to understand it, but really, I just couldn't get it. I read four books now, and some 6 pdf files and they don't give me a clear cut answer :( Alright, so this integral; ∫e-x2dx from -∞ to ∞, when converted to polar integral, limits become from 0 to 2∏ for the outer...

Homework Statement The larger context is that I'm looking at the scenario of fitting a polynomial to points with Gaussian errors using chi squared minimization. The point of this is to describe the likelihood of measuring a given parameter set from the fit. I'm taking N equally spaced x values...

Homework Statement I'm having difficulty solving the following integral. \int_{-\infty}^{\infty} x^{4}e^{-2\alpha x^{2}} \text{d}x Homework Equations \int_{-\infty}^{\infty} e^{-\alpha x^{2}} \text{d}x = \sqrt{\frac{\pi}{\alpha}} \int_{-\infty}^{\infty} x^{2}e^{-\alpha x^{2}}...

Hi folks, I'm trying to get from the established relation: $$ \int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} = a^{-2}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$ to the similarly derived: $$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = 3a^{-4} \int_{-\infty}^{\infty}...

Homework Statement Find the Gaussian integral: I = \int_{-\infty}^{\infty} e^{-x^2-4x-1}dx (That's all the information the task gives me, minus the I=, I just put it there to more easily show what I have tried to do) 2. The attempt at a solution I tried to square I and get a double...

How to show that the variance of the gaussian distribution using the probability function? I don't know how to solve for ∫r^2 Exp(-2r^2/2c^2) dr .

Homework Statement I'm re-hashing a problem from my notes; given the wave function \psi(x)=Ne^{-(x-x_0)/2k^2} Find the expectation value <x>. Homework Equations The normalization constant N for this is in my notes as N^2=1/\sqrt{2 \pi k^2} N=1/(2\pi k^2)^{(1/4)} It should be...

How do you do a gaussian integral when it contains a heaviside function!? Very few textbooks cover gaussian integrals effectively. This isn't a big deal as they are easy to locate in integral tables, but something I cannot find anywhere is how to handle a gaussian with a heaviside heaviside...

Hi everyone. The problem I have to face is to perform a taylor series expansion of the integral \int_{-\infty}^{\infty}\frac{e^{-\sum_{i}\frac{x_{i}^{2}}{2\epsilon}}}{\sqrt{2\pi\epsilon}^{N}}\cdot e^{f(\{x\})}dx_{i}\ldots dx_{N} with respect to variance \epsilon. I find some difficulties...

Hey everyone, I know, lots of threads and online information about Gaussian integrals. But still, I couldn't find what I am looking for: Is there a general formula for the integral \int_{\mathbb{R}^d} d^d y \left|\vec{y}\right| \exp(-\alpha \vec{y}^2) where y is a vector of arbitrary...

can some one tell me how to go about solving the gaussian integral e^(-x^2) I know it has no elementary integral . but i was told the improper integral from -inf to positive inf can be solved and some said that i haft to do it complex numbers or something and help would be great , this...

Integrating exp(x^2) like gaussian integral?? Hi, I can't solve this integral \int^{1}_{0}\\e^{x^2}\\dx Can I solve this integral like gaussian integral? Please help me Thanks.

Homework Statement Consider the gaussian distribution shown below \rho (x) = Ae^{-\lambda (x-a)^2 where A, a, and \lambda are positive real constants. Use \int^{-\infty}_{+\infty} \rho (x) \,dx = 1 to determine A. (Look up any integrals you need) Homework Equations Given in...

Homework Statement The time-evolution operator \hat{U}(t,t_0) for a time-dependent Hamiltonian can be expressed using a Magnus expansion, which can be written...