What is Gaussian integral: Definition and 65 Discussions

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function





{\displaystyle f(x)=e^{-x^{2}}}
over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is







{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.}
Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. 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.
Although no elementary function exists for the error function, as can be proven by the Risch algorithm, the Gaussian integral can be solved analytically through the methods of multivariable calculus. That is, there is no elementary indefinite integral for





{\displaystyle \int e^{-x^{2}}\,dx,}
but the definite integral





{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx}
can be evaluated. The definite integral of an arbitrary Gaussian function is








{\displaystyle \int _{-\infty }^{\infty }e^{-a(x+b)^{2}}\,dx={\sqrt {\frac {\pi }{a}}}.}

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  1. MevsEinstein

    B Is this identity containing the Gaussian Integral of any use?

    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?
  2. A

    MHB Multivariate gaussian integral

    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...
  3. murshid_islam

    I Gaussian integral by differentiating under the integral sign

    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...
  4. patric44

    Solving a Gaussian integral using a power series?

    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 ...
  5. B

    A Can this difficult Gaussian integral be done analytically?

    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...
  6. C

    MHB Inequality involving Gaussian integral

    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?
  7. looseleaf

    I Gaussian Integral Coordinate Change

    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...
  8. E

    Group delay with Gaussian pulse

    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...
  9. S

    I Gaussian integral in two dimensions

    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?
  10. S

    A Function integration of a Gaussian integral

    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\...
  11. Z

    Dealing w/slight modification of Gaussian integral?

    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...
  12. Adoniram

    Gaussian Integral: Converting from Cartesian to Polar

    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##...
  13. B

    How can the difficult Gaussian integral be solved using standard tricks?

    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...
  14. P

    Gaussian Integral with Denominator in QFT

    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...
  15. V

    Complex Gaussian Integral - Cauchy Integral Theorem

    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}...
  16. T

    Gaussian integral w/ imaginary coeff. in the exponential

    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...
  17. homer

    How Do You Compute the Complex Gaussian Integral I = ∫ e^(-ax^2 + ibx) dx?

    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...
  18. A

    Efficiently Solve Gaussian Integrals with our Homework Help Guide

    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...
  19. J

    How to Solve the Tough Gaussian Integral with a Constant in the Exponential?

    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...
  20. J

    What is the solution to the Gaussian integral?

    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...
  21. facenian

    Problem with Gaussian Integral

    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...
  22. skate_nerd

    Is there a formula for this gaussian integral

    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...
  23. skate_nerd

    MHB General formula for this weird Gaussian integral?

    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...
  24. S

    Solving Gaussian Integral: Stuck on Step

    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...
  25. M

    Gaussian integral using integration by parts

    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...
  26. S

    Gaussian Integral Simplification

    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)...
  27. U

    Integrating Difficult Gaussian Integrals for Multivariate Normal Distributions

    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}...
  28. F

    Integrating Gaussian integral by parts

    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...
  29. L

    Path integral and gaussian integral

    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...
  30. L

    Understanding Gaussian Integral: Question on Hinch's Perturbation Theory Book

    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...
  31. P

    Gaussian Integral Substitution

    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...
  32. F

    Average of Dirac Delta-Function over Double Gaussian Variables

    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...
  33. P

    Similar problem to Gaussian integral

    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?
  34. D

    Time development of a Gaussian integral help

    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'...
  35. A

    Why this relation is true when computing the Gaussian integral?

    \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...
  36. F

    Approximation of Gaussian integral arising in population genetics

    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...
  37. maistral

    Gaussian integral to polar coordinates - limit help?

    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...
  38. F

    Multidimensional Gaussian integral with constraints

    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...
  39. N

    Gaussian Integral: How to Solve for x^4 Term?

    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}}...
  40. A

    Why Does My Calculation of the Gaussian Integral for x^4 Differ?

    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}...
  41. N

    What is the method for solving the Gaussian integral?

    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...
  42. J

    Solving the Gaussian Integral for Variance of Gaussian Distribution

    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 .
  43. A

    Expectation Value/ Evaluating Gaussian Integral

    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...
  44. N

    How do you do a gaussian integral when it contains a heaviside function?

    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...
  45. N

    Taylor expansion of gaussian integral with respect to variance

    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...
  46. O

    Multidim. Gaussian integral with linear term

    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...
  47. C

    Solve Gaussian Integral: e^(-x^2)

    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...
  48. C

    Integrating exp(x^2) like gaussian integral?

    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.
  49. D

    Solve Gaussian Integral: A from \int^{-\infty}_{+\infty} \rho (x) \,dx = 1

    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...
  50. B

    QM: Magnus expansion, Gaussian integral

    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...