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
f
(
x
)
=
e
−
x
2
{\displaystyle f(x)=e^{-x^{2}}}
over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
∫
−
∞
∞
e
−
x
2
d
x
=
π
.
{\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
∫
e
−
x
2
d
x
,
{\displaystyle \int e^{-x^{2}}\,dx,}
but the definite integral
∫
−
∞
∞
e
−
x
2
d
x
{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx}
can be evaluated. The definite integral of an arbitrary Gaussian function is
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
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...
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...
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...