Theoretical analysis establishes an analytic framework for logarithmic complex numbers encoding the logarithm of zero, revealing rich geometric and cohomological structures.
We develop a complete analytic theory of logarithmic complex numbers, denoted by $L ∼=C[ε]/(ε2)$. This three-dimensional real algebra carries a natural interpretation in which the nilpotent direction ε = b encodes the logarithm of zero. We construct the topology, holomorphic function theory, contour integration, special functions (Gamma and Zeta), hybrid cohomology, and ordinary differential equations entirely within this framework. A central theme is the inevitability of the discontinuity of the collapse map E and the rich geometric structure that emerges from formalizing log(0) not as a single point, but as an entire compactified fiber. All proofs are given in full detail, and the exposition is self-contained.
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Jorge Vicente Romero (2026) studied this question.
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