Analytic Continuation of Divergent Integrals
Farhad Aghili
TL;DR
The paper addresses assigning finite meaning to the divergent monomial integral μ(s) = ∫_1^{∞} x^{-s} dx by analytic continuation, paralleling the Riemann ζ-function's continuation. It constructs a Dirichlet-series representation of μ(s) through term-by-term integration over unit intervals and Newton's binomial theorem, revealing a meromorphic link to ζ(s) and yielding a continuation to the full complex plane except for a simple pole at s = 1. The authors demonstrate that μ(s) can be written as μ(s) = (1/(s-1)) λ(s) with λ(s) → 1 on generic domains, and they carefully handle s in nonpositive integers to show cancellations of potential poles. As an application, they compute zero-point energy without a cutoff, obtaining ρ = (ħ c)/(16π^2), illustrating how analytic continuation provides a principled, cutoff-free finite value for otherwise divergent physical quantities.
Abstract
In this work, we investigate the improper integral of the monomial \(μ(s) = \int_1^{\infty} x^{-s} \,dx \) as a continuous analogue of the infinite series representation of the Riemann $ζ$-function, \(ζ(s) = \sum_{n=1}^{\infty} n^{-s}\). 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 $ζ$-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 \(μ\)-function and the \(ζ\)-function, leading to a functional equation that extends the divergent integral through analytic continuation and that the \(μ\)-function is holomorphic everywhere except at \(s = 1\).