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Gaussian functions?

  1. May 24, 2007 #1
    Is the summation of an infinite number of different (not just different by constants ) Gaussian functions still a gaussian function?

    I think not because you can becuase you can just pull an e^(ax^2) from the series where a is any value as small as needed.
     
    Last edited: May 24, 2007
  2. jcsd
  3. May 24, 2007 #2

    quasar987

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    You're asking if whatever A_n, mu_n and sigma_n,

    [tex]\sum_{n=1}^{\infty}A_n\exp{-\frac{(x-\mu_n)^2}{2\sigma_n^2}}=A\exp{-\frac{(x-\mu)^2}{2\sigma^2}}[/tex]

    for some A, mu and sigma. Is that correct?
     
  4. May 25, 2007 #3
    Yeah. Although I think this result dosen't hold?
     
  5. May 25, 2007 #4

    matt grime

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    If you sum two gaussians with mean X and Y (think of them as p.d.f.s) then you get one with mean X+Y. So an infinite sum will, in general, have 'infinite mean'. Similarly, infinite standard deviation. So, no, it is not a Gaussian, and in general won't even exist.
     
  6. May 29, 2007 #5
    But if you sum a finite number of different Gaussian than the sum will still be a gaussian? If so how? how can you get e^a+e^b=e^c where a,b,c are functions that satisfy the exponentials being a gaussian.
     
  7. May 29, 2007 #6

    matt grime

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    So, you're arguing that if property P holds for a finite number of things (terms, whatever), then it holds for an infinite number of things? Please, back off and think about that for a while. And I'm not arguing that e^a+e^b=e^c. What are a,b,c, for a start?
     
  8. May 29, 2007 #7
    What I am saying or rather asking is that I don't even see how it would work for the finite case. That is for any two different Gaussians
    http://en.wikipedia.org/wiki/Gaussian_function

    How do you add them and produce another gaussian?

    or even if you take e^2+e^3. Can you make it into e^a for any number a?
     
  9. May 29, 2007 #8
    Well, [itex]e^{2} + e^{3} = e^{a} \rightarrow a = ln(e^{2} + e^{3})[/itex].
     
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