What does this research mean for the field?

Evaluating arbitrary analytic functions using the field of formal Laurent series enables a three-dimensional visualization convention that maps classical outputs and their derivatives simultaneously, resolving structural singularities into clean geometric boundaries. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To introduce an algebraic and geometric framework that resolves singularities in complex analysis by projecting functions into a three-dimensional coordinate system.

August 29, 2026Open Access

The Zeta Axis: A Three-Dimensional Visualisation Convention for Arbitrary Complex Functions via the Field of Formal Laurent Series

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Authors

JRJorge Vicente Romero

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Implication

Mathematical framework demonstrates three-dimensional visualization of complex functions via formal Laurent series, uncovering continuous geometric representations of classical singularities.

Key Points

To introduce an algebraic and geometric framework that resolves singularities in complex analysis by projecting functions into a three-dimensional coordinate system.
Formulated an algebraic system treating zero as an invertible infinitesimal unit generating a third axis, grounded in the field of formal Laurent series.
Evaluated analytic functions at shifted inputs to simultaneously map function outputs and their first-order jets across real, imaginary, and zeta dimensions.
Constructed computational visualization routines in MATLAB to model continuous surfaces and pole relocations.
Transformed algebraic singularities and poles into smooth, structurally informative geometric boundaries in three dimensions.
Demonstrated exact geometric representations of classical functions, such as mapping the reciprocal function into a precise hyperbolic paraboloid.

Cite This Study

Jorge Vicente Romero (2026) studied this question.

synapsesocial.com/papers/6a9299c88e5d7d1fc0c12012 https://doi.org/10.5281/zenodo.22119143

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Implication

Mathematical framework demonstrates three-dimensional visualization of complex functions via formal Laurent series, uncovering continuous geometric representations of classical singularities.

The Zeta Axis: A Three-Dimensional Visualisation Convention for Arbitrary Complex Functions via the Field of Formal Laurent Series

Discussion

Member takes

Implication

Key Points

Cite This Study

The Zeta Axis: A Three-Dimensional Visualisation Convention for Arbitrary Complex Functions via the Field of Formal Laurent Series

Discussion

Member takes

Implication

Key Points

Cite This Study

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