Delta function could not be got when delta function is the answer

Discussion in 'Mathematica' started by PengYu.UT, Feb 11, 2006.

  1. PengYu.UT

    PengYu.UT Guest

    Sum[E^(-2*Pi*I*n*p*k), {n, -Infinity, Infinity}]

    The above summation should give an delta function. However, 0 is given
    by Mathematica 5.0. Is it a bug. Is there any workaround to get the
    delta function?

    Thanks,
    Peng
     
    PengYu.UT, Feb 11, 2006
    #1
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  2. PengYu.UT

    albert Guest

    which is no function but a distribution (or generalized function)...
    which is correct for almost every p and k :)
    depends on what you expect that mathematica can do. As far as I know, it can
    handle the delta function only to some extend in some functions like
    Integrate but does not claim to support distributions in general (see
    documentation for DiracDelta and 'Generalized Functions'). But maybe I am
    wrong here, I personally wouldn't expect mathematica to do too much useful
    stuff with DiracDelta especially if it is not in the realm of D, Integrate
    and FourierTransform...
    from the sum you used as input: I don't know.
    if input doesn't matter try these :):
    D[UnitStep[x], x]
    FourierTransform[1/Sqrt[2 Pi], k, x]

    hth,

    albert
     
    albert, Feb 12, 2006
    #2
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  3. PengYu.UT

    Pratik Desai Guest

    Check out the site
    http://functions.wolfram.com/GeneralizedFunctions/DiracDelta/
    and you can download a notebook with all kinds of info about DiracDelta,
    among which is this
    DiracDelta[x] == Sum[HoldForm[E^(I*k*x)], {k, -Infinity,
    Infinity}]/(2*Pi) /; -2*Pi < x < 2*Pi
    I added the holdform because mathematica automatically evaluates the sum
    to zero, I tried to use
    2.5.2 Manipulating Sets of Transformation Rules of m-book but was not
    able to apply the transformation??

    Hope this helps

    Pratik
     
    Pratik Desai, Feb 13, 2006
    #3
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