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Orthogonal Polynomials

  1. Nov 13, 2009 #1
    In my third year math class we were asked a question to prove that Ho(X) and H1(x) are orthogonal to H2(x), with respect to the weight function e^(-x^2) over the interval negative to positive infinity

    where Ho(x) = 1
    H1(x) = 2x
    H2(x) = (4x^2) - 2

    i know that i have to multiply Ho(x) by H2(x) and divide by the weight function and integrate..but i get lost when it comes to integrating by parts with e^(x^2)...
     
  2. jcsd
  3. Nov 13, 2009 #2
    in this question Hn(x) is the herite polynomial...where n = 0, 1, 2 ,3 etc
     
  4. Nov 13, 2009 #3

    LCKurtz

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    Since you have symmetry in x you can do this integral:

    [tex]2\int_0^\infty (4x^2-2)e^{-x^2}\,dx[/tex]

    Try breaking it into two parts and on the first part use integration by parts with:

    [tex]u = 2x\ dv = 2xe^{-x^2}dx[/tex]

    and if I'm not mistaken, nice things will happen.
     
  5. Nov 13, 2009 #4
    well that looks right..but it should be e^(x^2)...
     
  6. Nov 13, 2009 #5

    LCKurtz

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    Why do you say that? The integral won't even converge with a positive exponential in there.
     
  7. Nov 13, 2009 #6
    you are supposed to divide by the weight function...which is e^(-x^2)
     
  8. Nov 13, 2009 #7

    LCKurtz

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    No you aren't. And like I said, the integral wouldn't converge.
     
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