1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Unique properties of Gaussians

  1. Apr 22, 2010 #1
    Hello,
    let's suppose we are given a Gaussian function [tex]f(x)=Ae^{-ax^2}[/tex] (where a,A are real scalars and a is positive)

    Is it possible to (dis)prove that the following identity is true only for Gaussians?

    [tex]\frac{\partial^2}{\partial x^2} log f(x) = const[/tex]

    Thanks!
     
    Last edited: Apr 22, 2010
  2. jcsd
  3. Apr 22, 2010 #2
    If you solve the differential equation you end up with
    [tex] f(x) = A e^{const\. x^2 + B x}.[/tex]
    So provided the constant is negative, you get a gaussian with a shift in the variable.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook