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
Homework Statement
Basically, i have to find the solution to:
Int( x2 * exp (-(x-w)^2) , x= -infinity.. infinity)
Homework Equations
I realize this is connected to Gaussian Integration. So that if i have Int (exp(-x2), x=-infinity ... infinity) the answer is sqrt(Pi)
Also, i...
Homework Statement
Solve:
In = \int_{0}^{\infty} x^n e^{-\lambda x^2} dx
Homework Equations
The Attempt at a Solution
So my teacher gave a few hints regarding this. She first said to evaluate when n = 0, then consider the cases when n = even and n = odd, comparing the even...
EDIT: meant to post this is the math forums, if you can remove this I'm going to switch it over
Homework Statement
Solve:
In = \int_{0}^{\infty} x^n e^{-\lambda x^2} dx Homework Equations
The Attempt at a Solution
So my teacher gave a few hints regarding this. She first said to evaluate when...
Hi,
I read the chapter "Anticommuting Numbers" by Peskin & Schröder (page 299) about Grassmann Numbers and now I would like to prove
\int d \bar{\theta}_1 d \theta_1 ... d \bar{\theta}_N d \theta_N e^{-\bar{\theta} A \theta} = det A
\theta_i are complex Grassmann Numbers...
I see that the formula for this general integral is
\int^{+\infty}_{-\infty} x^{2}e^{-Ax^{2}}dx=\frac{\sqrt{\pi}}{2A^{3/2}}
However, I am not getting this form with my function. I transformed the integral using integration by parts so that I could use another gaussian integral that I knew at...
I have the following Gaussian Integral:
\int_{0}^{\infty}2\pi r\left |{\int_{-l/2}^{l/2}\frac{e^\frac{-r^2}{bH}}{(1+ix)(k^{''} - ik^{'}x)H}}dx\right |^2dr
Where
H = \frac{1+x^2}{k^{''} - ik^{'}x} - i\frac{x - \zeta}{k^{'}}
Assume any characters not defined are constants.
I agree...
Homework Statement
We know that
\int_{-\infty}^\infty e^{-ax^2}dx = \sqrt{\pi \over a}.
Does this hold even if a is complex?
Homework Equations
The Attempt at a Solution
In the derivation of the above equation, I don't see any reason why we must assume that a be real. So I...
Homework Statement
Given f(x) = e^{-ax^2/2} with a > 0 then show that \^{f} = \int_{-\infty}^{\infty} e^{-i \xi x - ax^2/2} \, \mathrm{d}x = \surd\frac{2}{a} = e^{-\xi^2/2a} by completing the square in the exponent, using Cauchy's theorem to shift the path of integration from the real axis...
OK so we have:
\int f(z) e^{a g(z)} dz^3
integerated over all space.
Now there is a identity for this integral as an average, or something like that, right? What is it? Or perhaps you have suggestions where I could read up on that kind of thing?
(I'm not looking for the integral in...
Hello,.. that's part of a problem i find in QFT (i won't explain it since it can be very tedious), the question is that i must evaluate the Multi-dimensional Gaussian Integral.
\int_{-\infty}^{\infty}d^{n}V exp(x^{T}Ax)exp(ag(x))
for n\rightarrow \infty of course if the integral is...
Hi,
I'm trying to evaluate the standard Gaussian integral
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
The standard method seems to be by i)squaring the integral, ii)then by setting the product of the two integrals equal to the iterated integral constructed by composing the two...
I am trying to to the Gaussian integral using contour integration.
What terrible mistake have I made.
I = \int_{-\infty}^\infty \mathrm{e}^{-x^2} \mathrm{d}x
I consider the following integral:
\int_C \mathrm{e}^{-z^2} \mathrm{d}z
where C is the half-circle (of infinite...
Hey, I've been learning about gaussian integrals lately. And I'm now stuck in one part. I am now trying to derive some kind of general formula for gaussian integrals
\int x^n e^{-\alpha x^2}
for the case where n is even. So they ask me to evaluate the special case n=0 and alpha=1. So its...