Dimensional Analysis and Complex Analysis: An Analogy via Singularity Types

This paper establishes a conceptual correspondence between dimensional analysis in physics and singularity analysis in complex analysis. We demonstrate that translating dimensional analysis into the language of complex analysis fundamentally involves classifying singularity types. Dimensional analysis, which governs physical units and dimensionless combinations, is shown to exhibit deep structural parallels with the behavior of complex functions near singular points (poles, zeros, essential singularities, branch points). Key analogies include: (1) Physical divergences/zeros correspond to poles/zeros in complex functions, (2) Dimensionless parameters (e.g., Reynolds number) govern physical regimes similarly to Laurent series exponents controlling singularity behavior, (3) Dimensional homogeneity mirrors analyticity requirements, and (4) The rank of dimensional matrices corresponds to the order of singularities. Through examples and theoretical analysis, we prove that dimensional inconsistencies represent "non-analytic" points in physical descriptions, while dimensionless groups act as singularity classifiers. This unification provides new insights into scaling laws and critical phenomena.