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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.
     
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