# Zeta regularization and product of dirac delta distribution

1. Jun 27, 2011

### zetafunction

using the convolution theorem with power functions $$x^{m}$$ we may define via the convolution theorem the product of 2 dirac delta distribution

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

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

$$\int_{R}dt t^{m}$$ (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

$$\int_{R}dt(x-t)^{m}t^{n}$$

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 $$(m+1)\int_{0}^{N}x^{m}dx=N^{m+1}$$