Gabriel Ben-Simon | Searcharxiv

arXiv2026

Cauchy-Type Convolution Integrals as the Starting Point for a First Encounter with Complex Analysis

Alongside three established presentations of elementary complex analysis, we propose an approach centered on one organizing question: when is contour convolution well defined on homotopy classes? From this starting point, we develop the foundational results of the subject. A supporting conceptual distinction identifies those elements of real and complex analysis that share the same operational and symbolic rules. We refer to this collection as the "Toolbox." Examples and structural connections illustrate the explanatory value and reach of the approach across different aspects of the theory. We also examine the relationship of our approach to the established presentations.

math.CV↗

arXiv2026

Exactness as an Explanatory and Computational Target: A Reduction Architecture for First-Order ODEs, with a Higher-Order Outlook

A typical first-order ODE syllabus is often presented as a sequence of named methods, which can obscure the unity among them. We propose organizing a substantial part of the classical symbolic first-order syllabus around a single computational target: an exact representative. For a regular scalar equation represented by a one-form $ω$, the reduction problem is to find, on an appropriate coordinate patch, a coordinate change $Φ$ and a nonvanishing factor $μ$ such that $μΦ^*ω=dG$, so that solution curves are level sets $G=C$. This yields a two-floor architecture. The structural floor records the target, transformations, and domain restrictions; the operational floor executes substitutions, normalizations, and integrations. Linear and separable equations arise from the simplest one-variable integrating factors, while homogeneous, Bernoulli, and related classes become reduction paths to the same target. In an AI- and CAS-assisted environment, this shifts emphasis toward recognition, justification, auditing, and interpretation. The extended version also develops reverse construction, a structural exercise laboratory, comparisons of reduction paths, and a higher-order outlook through first integrals, operator factorization, and structural normalization.

math.GM↗