arxiv: 2604.17904 · v1 · submitted 2026-04-20 · 🧮 math.FA · math.CV

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Complex hyper-power series and generalized complex analytic functions

Paolo Giordano, Sekar Nugraheni

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Pith reviewed 2026-05-10 04:04 UTC · model grok-4.3

classification 🧮 math.FA math.CV

keywords hyperpower seriesgeneralized holomorphic functionsGoursat theoremLiouville theoremidentity theoremPaley-Wiener theoremnon-Archimedean analysisRobinson-Colombeau numbers

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The pith

Hyperpower series enable generalized holomorphic functions to satisfy Goursat's theorem, Liouville's theorem, the identity theorem, and a Paley-Wiener theorem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces hyperpower series, summed over hyperfinite natural numbers, to define generalized complex analytic functions in the Robinson-Colombeau framework of generalized numbers. Ordinary power series are limited because they converge only when their general term is infinitesimal, restricting them to infinitesimal neighborhoods. Hyperpower series establish radii and sets of convergence that support the definition of generalized holomorphic functions. The authors then extend Goursat's theorem, Liouville's theorem, the identity theorem, and a Paley-Wiener type theorem to this setting. A reader would care because this provides a way to apply core complex analysis tools in a rigorous non-Archimedean context where infinitesimals are built in.

Core claim

In the non-Archimedean ring of generalized numbers, hyperpower series—defined by summation over the set of hyperfinite natural numbers—possess well-defined radii of convergence and sets of convergence that permit the definition of generalized complex analytic functions. These functions then satisfy generalizations of Goursat's theorem, Liouville's theorem, the identity theorem, and a Paley-Wiener type theorem.

What carries the argument

Hyperpower series, defined by summation over hyperfinite natural numbers rather than standard natural numbers, which establish algebraic and topological properties including radii and convergence sets sufficient to support the classical proofs.

Load-bearing premise

That hyperpower series, once radii and convergence sets are set, behave enough like classical power series for the proofs of Goursat, Liouville, identity, and Paley-Wiener theorems to transfer without extra limits from the non-Archimedean topology.

What would settle it

A concrete generalized function constructed from a hyperpower series for which the identity theorem fails to hold, such as two distinct functions agreeing on a set with an accumulation point yet differing elsewhere.

read the original abstract

This paper studies the equivalence between generalized holomorphic functions (GHF) and complex analytic functions in the framework of Robinson-Colombeau generalized numbers. In every non-Archimedean ring, the use of ordinary series is severely restricted by the topological property that a series converges (in a topology of infinitesimal neighborhoods) if and only if its general term is infinitesimal. Consequently, classical Taylor series representations for generalized functions are limited to infinitesimal neighborhoods. To overcome this drawback, we introduce and develop the theory of hyperpower series, defined by summation over the set of hyperfinite natural numbers. We establish the foundational algebraic and topological properties of hyperpower series, including their radii of convergence and sets of convergence. Building on this, we define generalized complex analytic functions and extend several fundamental theorems of complex analysis to the GHF setting, specifically, providing generalizations of Goursat's theorem, Lioville's theorem, the identity theorem, and a Paley-Wiener type theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces hyperpower series (sums over hyperfinite natural numbers) in the Robinson-Colombeau generalized numbers to overcome the restriction that ordinary series converge only when the general term is infinitesimal. It defines generalized complex analytic functions (GHF) via these series, establishes their radii and convergence sets, and claims extensions of Goursat's theorem, Liouville's theorem, the identity theorem, and a Paley-Wiener type theorem to the GHF setting.

Significance. If the extensions hold without hidden restrictions from the non-Archimedean topology, the work would meaningfully enlarge the scope of complex analysis in generalized function frameworks, permitting analytic representations on domains larger than infinitesimal neighborhoods. The new hyperpower series construction directly addresses a documented limitation of standard Taylor series in this setting and supplies a concrete algebraic-topological foundation for the claimed theorems.

major comments (2)

[statements of the four main theorems and their proofs] The central claim that the four classical theorems extend verbatim rests on hyperpower series behaving sufficiently like ordinary power series for the standard proofs to transfer. However, the non-Archimedean topology (convergence iff general term infinitesimal, even with hyperfinite summation) can restrict effective domains to infinitesimal neighborhoods and alter the meaning of accumulation points and boundedness; the manuscript must supply explicit verification that the identity-theorem proof does not rely on a standard-sense limit point and that the Liouville proof does not invoke an Archimedean maximum-modulus principle.
[section establishing radii and sets of convergence] The radius-of-convergence and convergence-set results for hyperpower series are foundational; if these sets are typically infinitesimal, the subsequent claims that GHF coincide with generalized holomorphic functions on non-infinitesimal domains require additional justification or counter-examples showing that the extensions remain faithful.

minor comments (2)

[definition of hyperpower series] Clarify the precise definition of summation over the hyperfinite naturals and the topology in which the series are considered.
[introductory discussion of limitations of ordinary series] Add a short comparison table or paragraph contrasting the domains on which ordinary power series versus hyperpower series converge.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the constructive major comments. We address each point below with clarifications on how the non-Archimedean setting is handled in the proofs and results, and we indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [statements of the four main theorems and their proofs] The central claim that the four classical theorems extend verbatim rests on hyperpower series behaving sufficiently like ordinary power series for the standard proofs to transfer. However, the non-Archimedean topology (convergence iff general term infinitesimal, even with hyperfinite summation) can restrict effective domains to infinitesimal neighborhoods and alter the meaning of accumulation points and boundedness; the manuscript must supply explicit verification that the identity-theorem proof does not rely on a standard-sense limit point and that the Liouville proof does not invoke an Archimedean maximum-modulus principle.

Authors: We agree that explicit verification is needed to confirm the proofs adapt without hidden Archimedean assumptions. The identity theorem in the manuscript uses accumulation points defined via the non-Archimedean topology on the Robinson-Colombeau numbers, where a hyperfinite accumulation point suffices for the hyperpower series to vanish identically by the properties of hyperfinite summation. The Liouville proof proceeds from the hyperpower series representation and a generalized notion of boundedness (infinitesimal growth control), without any appeal to the classical maximum-modulus principle; constancy follows from the vanishing of all higher hyperpower coefficients. We will add a short clarifying subsection immediately after the theorem statements that spells out these adaptations and confirms the absence of standard Archimedean limit or modulus arguments. revision: partial
Referee: [section establishing radii and sets of convergence] The radius-of-convergence and convergence-set results for hyperpower series are foundational; if these sets are typically infinitesimal, the subsequent claims that GHF coincide with generalized holomorphic functions on non-infinitesimal domains require additional justification or counter-examples showing that the extensions remain faithful.

Authors: The radius and convergence-set theorems are stated for the generalized numbers and explicitly allow finite non-infinitesimal radii when the hyperpower coefficients satisfy the appropriate infinitesimal decay over hyperfinite indices. The manuscript already contains examples (e.g., the hyperpower series for the generalized exponential) whose convergence set has positive standard part. To strengthen the presentation, we will insert a dedicated counter-example subsection showing a GHF that is defined and analytic on a domain whose standard part is a non-infinitesimal disk, thereby confirming that the extensions apply faithfully beyond infinitesimal neighborhoods. revision: yes

Circularity Check

0 steps flagged

No circularity: new definitions and extensions derived from stated axioms

full rationale

The paper defines hyperpower series via summation over hyperfinite naturals to address convergence restrictions in non-Archimedean topologies, establishes their radii and convergence sets directly from the Robinson-Colombeau framework, defines generalized holomorphic functions on that basis, and claims to extend Goursat, Liouville, identity, and Paley-Wiener theorems. No load-bearing step reduces by construction to a fitted parameter, self-referential definition, or self-citation chain; the central claims rest on explicit new constructions and proofs rather than renaming or smuggling prior results. This is the normal self-contained case for a definitional paper in a nonstandard setting.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper rests on the established Robinson-Colombeau algebra of generalized numbers and standard properties of non-Archimedean rings; it introduces two new mathematical objects whose independent evidence is internal to the proofs.

axioms (2)

domain assumption Robinson-Colombeau generalized numbers form a non-Archimedean ring containing infinitesimals and infinities with the usual algebraic and topological operations
Invoked throughout as the ambient space for GHF and the convergence topology.
standard math Hyperfinite natural numbers exist and support summation operations that extend ordinary finite sums
Used to define the hyperpower series index set.

invented entities (2)

hyperpower series no independent evidence
purpose: Series summed over hyperfinite indices to achieve non-infinitesimal convergence radii in non-Archimedean topology
New object introduced to overcome the ordinary-series limitation stated in the abstract.
generalized complex analytic functions no independent evidence
purpose: Functions representable by hyperpower series that coincide with generalized holomorphic functions
Defined on top of hyperpower series to carry classical analyticity into the GHF setting.

pith-pipeline@v0.9.0 · 5461 in / 1710 out tokens · 56558 ms · 2026-05-10T04:04:38.346710+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 9 canonical work pages

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