Gaussian integral Definition and 62 Threads

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...

How can I evaluate something like \int^{\infty}_{-\infty} d^3 \mathbf{x} f(\mathbf{x}) e^{t g(\mathbf{x})} in Mathematica, where x is a vector in 3D?

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...

Evaluate: \frac{1}{\sqrt{2\pi} \sigma} \int_{-\infty}^{\infty} \, dx \, exp\left[-\frac{(x - \mu)^2}{2\sigma^2}\right] \, , where $\mu$ and $\sigma$ are complex numbers. I tried writing \begin{align} \sigma &= s_1 + is_2 \,\\ \mu &= m_1 + i m_2 \, . \end{align} The integral...

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...