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Zeta regularization and product of dirac delta distribution

  1. Jun 27, 2011 #1
    using the convolution theorem with power functions [tex] x^{m} [/tex] we may define via the convolution theorem the product of 2 dirac delta distribution

    then main idea is to consider the convolution integral [tex] \int_{R}dt(x-t)^{m}t^{n} [/tex]

    and then apply the Fourier transform with respect to variable 'x' (here , and n are positive integers)

    of course, this integral will be DIVERGENT for any value of 'x' due to the expressions

    [tex] \int_{R}dt t^{m} [/tex] (integrals over the Real line)

    however by making an extension of the Zeta regularization algorithm for divergent series , we can make sense of integrals of the form

    [tex] \int_{R}dt(x-t)^{m}t^{n} [/tex]

    here.. http://vixra.org/abs/1005.0071

    we present some examples of product of distributions involving the dirac delta function, Heaviside Step function and finite part (in Cauchy's sense) , we discuss some of the applications of the Zeta regularization algorithm for integrals and how in the limit of finite N (upper limit) we get the expected result of calculus [tex] (m+1)\int_{0}^{N}x^{m}dx=N^{m+1} [/tex]
     
  2. jcsd
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