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Find the following integral

  1. Aug 20, 2007 #1
    can anybody find the result for the following equation:

    F(d)= [tex]\int^{T_f}_{0}p(t)p(t-d) dt [/tex]

    where
    916; = d but it doesnt appears very well
    and
    p(t) = (-1)^n * e^(t^2) * d/dt * e^(-t^2)

    thanks alot!
     
    Last edited: Aug 20, 2007
  2. jcsd
  3. Aug 20, 2007 #2

    dextercioby

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    The way you've written, it appears that

    [tex] p(t)=-2 (-1)^n t [/tex]

    I believe that in this case the integral would be a formality.
     
  4. Aug 20, 2007 #3
    I am sorry I didnt write it correctly

    p(t) = (-1)^n * e^(t^2) * d^n/dt^n * e^(-t^2)

    where n= 1,2,...,N

    I just want a general formula for the result
     
  5. Aug 20, 2007 #4

    EnumaElish

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    Have you attempted at an answer? What technique(s) have you tried? Integration by parts comes to mind...
     
  6. Aug 21, 2007 #5
    my question is how to find a general formula for the following

    F(d)= [tex]\int^{T_f}_{0}p(t)p(t-d) dt [/tex]

    where
    p(t) = (-1)^n * e^(t^2) * d/dt * e^(-t^2)

    and
    n=1,2,...,N
    Thanks alot!
     
  7. Aug 21, 2007 #6

    EnumaElish

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  8. Aug 21, 2007 #7

    matt grime

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    And my response is 'why do you keep putting n's in and then taking them out?' State the question precisely. I presume you're just supposed to do it for n an integer, rather than for n=1,2,3,...
     
  9. Aug 21, 2007 #8

    dextercioby

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    That p_{n} is, up to a normalization constant, a Hermite polynomial of degree "n". So try looking that integral in mathematical tables of integrals or ask Mathematica software about the result.
     
  10. Aug 23, 2007 #9

    EnumaElish

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    "There are many functions - called special functions - which fail to
    have an anti-derivative expressible as a finite combination of
    elementary functions. The so-called elliptic functions, the error
    function, and the gamma function are a few examples. The error
    function, which is extremely useful in both physics and statistics, is
    defined as:

    erf(x) = (2/sqrt(pi))integral from 0 to x of e^(-t^2)dt

    Extensive tables of the error function would not exist if the
    anti-derivative of e^(-t^2) were expressible as a finite combination
    of elementary functions."
    http://mathforum.org/library/drmath/view/53554.html
     
  11. Aug 23, 2007 #10

    Dick

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    p(t) is not an exponential. It's a polynomial as dextercioby has already pointed out.
     
  12. Aug 23, 2007 #11

    EnumaElish

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    I have realized the same because, as dextercioby had pointed out, there is a closed solution for n=1.

    Here's how I'd approach the problem (and I communicated this to T.Engineer at least once before, under Statistics & Probability). I'd start with n=1 and calculate the closed solution, which is easy. Then move on to n=2, 3, ..., and see if there is a pattern.
     
  13. Aug 23, 2007 #12

    Dick

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    There are also recurrence relations for the Hermite polynomials which could be exploited without rediscovering them empirically. It's clear there is no exponential in the final integral since exp(-t^2) comes out from the differentiation intact and cancels with the exp(t^2) without even knowing there are closed form solutions for particular n.
     
  14. Aug 24, 2007 #13

    EnumaElish

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    Right; the fact that a closed solution exists is sufficient but not necessary to see that it is not the error function, or any other special function.
     
  15. Aug 24, 2007 #14

    Dick

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    If anybody is interested in pursuing this, here's a Maxima (free math software) program to compute the nth case. It should be pretty self explanatory.

    p(t,n):=expand(exp(t^2)*(-1)^n*diff(exp(-t^2),t,n));
    integ(n):=p(t,n)*subst((t-d),t,p(t,n));
    final(n):=integrate(integ(n),t,0,Tf);

    E.g. typing 'final(7);' after this is entered computes the n=7 case.
     
    Last edited: Aug 24, 2007
  16. Aug 24, 2007 #15
    Please, can you tell me from where can I download this program.
    Thanks a lot!
     
  17. Aug 24, 2007 #16

    EnumaElish

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  18. Aug 24, 2007 #17

    Dick

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    Try http://maxima.sourceforge.net/. If you are running a debian flavor linux it might be as simple doing 'apt-get install xmaxima'.
     
  19. Aug 24, 2007 #18
    I am runing windows Xp.
     
  20. Aug 24, 2007 #19

    Dick

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    Then get, for example, maxima-5.12.0a.exe from the downloads section.
     
  21. Sep 3, 2007 #20

    can you help to find a general formula for the autocorrelation function Hermite polynomials.
    Thanks a lot!
     
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