Analytic Continuation of Divergent Integrals

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 $s = 1$.