For $A$ a commutative ring and $p$ a prime number, then there is the topological completion at $(p)$, the p-completion $A_p^\wedge \coloneqq \underset{\leftarrow}{\lim}_n A/(p^n A)$. The analytic completion is instead the quotient
For $A= \mathbb{Z}$ the integers both constructions agree, up to isomorphism, and yield the p-adic integers. In general though they are different.
Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol 304, Springer 1972
Charles Rezk, Analytic completion (pdf)
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