Zeta regularization and product of dirac delta distribution

In summary, the convolution theorem is used to define the product of two Dirac Delta distributions using power functions x^{m}. The main idea is to consider the convolution integral \int_{R}dt(x-t)^{m}t^{n} and apply the Fourier transform with respect to variable 'x'. However, this integral is divergent for any value of 'x', but by using the Zeta regularization algorithm, we can make sense of it. Examples of products of distributions involving the Dirac Delta function and applications of the Zeta regularization algorithm are also presented, with the expected result of calculus being obtained in the limit of finite N (upper limit). Finally, the result of the convolution theorem shows that the product of the Dirac
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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]
 
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/ (m+1) in this way we can make sense of the product of the dirac delta distribution and the power function x^{m} , as a result of the convolution theorem the result is that the product of the Dirac Delta Function and the power function x^{m} is equivalent to the finite part of the integral \int_{R}dt(x-t)^{m}t^{n} as obtained via Zeta regularization algorithm
 

Related to Zeta regularization and product of dirac delta distribution

1. What is Zeta regularization?

Zeta regularization is a mathematical technique used to assign values to divergent series or integrals that would otherwise be undefined. It involves using the Riemann zeta function to regularize the series or integral and give it a finite value.

2. How does Zeta regularization work?

Zeta regularization works by using the Riemann zeta function to assign a value to the divergent series or integral. The zeta function is a special function in mathematics that can be used to evaluate infinite series or integrals. It involves summing the terms of the series or integral in a specific way to obtain a finite value.

3. What is the product of Dirac delta distribution?

The product of Dirac delta distribution is a mathematical concept that arises when multiplying two or more Dirac delta distributions together. It is a way to represent the distribution of a product of two or more random variables, and it can be used to solve certain types of integrals.

4. How is Zeta regularization related to the product of Dirac delta distribution?

Zeta regularization and the product of Dirac delta distribution are related in the sense that they both involve manipulating divergent series or integrals to give them a finite value. Zeta regularization can be used to evaluate the product of Dirac delta distribution in certain cases, as it involves evaluating infinite series or integrals.

5. What are some applications of Zeta regularization and the product of Dirac delta distribution?

Zeta regularization and the product of Dirac delta distribution have various applications in physics, engineering, and mathematics. For example, they are used in quantum field theory, statistical mechanics, and signal processing. They can also be applied in solving certain types of differential equations and calculating certain types of integrals.

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