Analytic Continuation of Divergent Integrals (2110.06272v2)

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).