Analytical Solution to Improper Integral of Divergent Power Functions Using The Riemann Zeta Function

TL;DR

This paper derives an analytical solution for the improper integral of divergent power functions using the Riemann zeta function, revealing different results for integer and non-integer exponents.

Contribution

It introduces a novel method to evaluate improper integrals of power functions by expressing them in terms of Riemann zeta functions and Bernoulli numbers, depending on the nature of the exponent.

Findings

For integer exponents, the integral simplifies to a rational function.

For non-integer exponents, the integral evaluates to zero.

The solution connects improper integrals with special functions like zeta and Bernoulli numbers.

Abstract

This paper presents an analytical closed-form solution to improper integral $\mu(r)=\int_0^{\infty} x^r dx$, where $r \geq 0$. The solution technique is based on splitting the improper integral into an infinite sum of definite integrals with successive integer limits. The exact solution of every definite integral is obtained by making use of the binomial polynomial expansion, which then allows expression of the entire summation equivalently in terms of a weighted sum of Riemann zeta functions. It turns out that the solution fundamentally depends on whether or not $r$ is an integer. If $r$ is a non-negative integer, then the solution is manifested in a finite series of weighted Bernoulli numbers, which is then drastically simplified to a second order rational function $\mu(r)=(-1)^{r+1}/(r+1)(r+2)$. This is achieved by taking advantage of the relationships between Bernoulli numbers and binomial coefficients. On the other hand, if $r$ is a non-integer real-valued number, then we prove $\mu(r)=0$ by the virtue of the elegant relationships between zeta and gamma functions and their properties.