Divergent Integrals of Certain Analytical Functions In the Sense of Zeta Regularization

TL;DR

This paper extends Zeta function regularization to assign finite values to divergent integrals of analytical transcendental functions, providing a new formula that avoids cut-off functions and is applicable to various functions.

Contribution

The paper introduces a novel Zeta regularization method for divergent integrals of analytical functions, deriving a convergent series representation without cut-off functions.

Findings

Derived a new series formula for divergent integrals of analytical functions.

Demonstrated the method on several transcendental functions.

Provided a convergent series under specific conditions.

Abstract

In this paper we extend the Zeta function regularization technique, which gives a meaningful solution to divergent power series, in order to assign finite values to divergent integral of certain transcendental functions $f(x)$. The functions are assumed to be analytical and hence they have a convergent Maclaurin series with infinity radius of convergence. Using Maclaurin series and binomial expansion, we equivalently convert the divergent integral to an infinite series in terms of Riemann zeta function. It is shown that the infinite series is convergent under a certain condition and consequently the solution of the divergent integral becomes $\int_0^{\infty} f(x) dx = \sum_{k=0}^{\infty} \frac{(-1)^{k+1}}{(k+2)!} f^{(k)}(0)$. The advantage of this Zeta regularization technique is that it does not not require introducing any cut-off function to calculate such divergent integrals. This formula is applied to calculate the divergent integral of several common transcendental functions.