Analytic Continuation of Divergent Integrals

Abstract

In this work, we investigate the improper integral of the monomial \(\mu(s) = \int_1^{\infty} x^{-s} \,dx \) as a continuous analogue of the infinite series representation of the Riemann $\zeta$-function, \(\zeta(s) = \sum_{n=1}^{\infty} n^{-s}\). Both the monomial integral and the corresponding series converge for \(\mathrm{Re}(s) > 1\) and diverge for \(s \in \mathbb{C}\) with \(\mathrm{Re}(s) \leq 1\). In this paper, we construct an analytic continuation of the divergent monomial integral to the entire complex plane, excluding a simple pole at \(s = 1\), mirroring the analytic continuation of the $\zeta$-function. By performing term-by-term integration of the monomial over successive integer intervals and leveraging Newton's generalization of the binomial theorem, we express the improper integral as a Dirichlet series. This approach establishes an elegant relationship between the \(\mu\)-function and the \(\zeta\)-function, leading to a functional equation that extends the divergent integral through analytic continuation and that the \(\mu\)-function is holomorphic everywhere except at \(s = 1\).