# Universal Analysis An extension of classical calculus and function theory to the universal number domain, providing a unified framework for analytical methods across different number systems. Universal Analysis extends classical calculus and function theory to the universal number domain, providing a unified framework for analytical methods across different number systems while maintaining complete coherence with UOR principles. Universal analysis begins with a generalized concept of differentiation that unifies complex and p-adic approaches. For a function f: 𝕌 → 𝕌, the universal derivative at a point α ∈ 𝕌 is defined as f'(α) = lim(h→0) [f(α+h) - f(α)]/h where the limit is taken with respect to the universal metric. A function f: 𝕌 → 𝕌 is universally differentiable at α if and only if it is complex-analytically differentiable with respect to the complex embedding, it is p-adically differentiable with respect to every p-adic embedding, and the derivatives from these different perspectives coincide under the canonical embeddings. If a function f admits a prime-coordinate representation φ(f(η)) = F(φ(η)) for some coordinate function F, then the universal derivative can be computed through φ(f'(η)) = JF(φ(η)) where JF is the Jacobian of F in the coordinate space. Universal integration unifies complex and p-adic integration. For a function f: 𝕌 → 𝕌 and a path γ, the universal line integral is defined as ∫_γ f(η) dη which simultaneously represents a complex contour integral and a collection of p-adic integrals. If f: 𝕌 → 𝕌 is a universally differentiable function and γ is a path from a to b, then ∫_γ f'(η) dη = f(b) - f(a). This theorem generalizes both the Fundamental Theorem of Calculus and its p-adic analogues. A function f: 𝕌 → 𝕌 is universally analytic if it can be represented locally by a power series f(η) = ∑ a_n (η - α)^n that converges in both the Archimedean and all non-Archimedean metrics. Every universally analytic function f admits a Taylor expansion around any point α ∈ 𝕌: f(η) = ∑ f^(n)(α)/n! · (η - α)^n where the convergence is uniform in the universal metric within the domain of analyticity. Universal analysis supports rich function spaces, including the space L²(𝕌) of square-integrable universal functions which forms a Hilbert space. Every function f ∈ L²(𝕌) admits a generalized Fourier expansion in terms of universal characters. For a linear differential equation with universal coefficients, there exists a unique universally analytic solution in a neighborhood of any non-singular point, and this solution can be computed effectively through prime-coordinate algorithms. Universal analysis establishes profound connections between complex and p-adic analysis through theorems like the Universal Residue Theorem and Complex-p-adic Duality. The theory of universal analysis satisfies coherence axioms that align with UOR principles, including Analytical Extension, Coordinate Analyticity, Computational Effectiveness, and Observer Invariance. ## References - [[uor-c-050|universal-derivative]] - [[uor-c-051|universal-integration]] - [[uor-c-052|universal-analytic-functions]] - [[uor-c-053|universal-analysis-axioms]] ## Metadata - **ID:** urn:uor:resource:universal-analysis - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z