Analytic Continuation of Generalized Trigonometric Functions

Pisheng Ding

Analytic Continuation of Generalized Trigonometric Functions

T0 review · 0 major / 1 minor · reviewed 2026-05-25 · grok-4.3

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Pith's one-line read Generalized trigonometric functions with two parameters extend analytically to maximal domains of univalence via a geometric approach.

desk verdict The paper uses a geometric construction to extend a two-parameter family of generalized trigonometric functions to their maximal univalent domains and derives some standard consequences from that. read the letter →

arxiv 2106.15109 v4 pith:RJHIDVXB submitted 2021-06-29 math.CV

Pisheng Ding This is my paper

classification math.CV

keywords generalizedtrigonometricfunctionsanalyticcontinuationunivalentdomainsgeometricapproachMaclaurinseriesperiodicitycomplexanalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that a class of generalized trigonometric functions depending on two parameters can be analytically continued to the largest domains on which they remain univalent. It does so by means of a single geometric construction that replaces case-by-case arguments. A reader would care because univalence guarantees local invertibility, which controls the global mapping properties of the functions and the radius at which their power series converge. The same construction also produces explicit information on rotational symmetry, continuation past the univalent region, and periodicity.

What carries the argument

The unified geometric representation of the two-parameter generalized trigonometric functions that directly supplies their maximal univalent domains.

What would settle it

An explicit point inside one of the claimed maximal domains at which the function fails to be univalent, or at which analytic continuation is impossible.

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Extended reading notes

Core claim

Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning radius of convergence for the Maclaurin series, commutation with rotation, continuation beyond the domain of univalence, and periodicity.

Load-bearing premise

The two-parameter generalized trigonometric functions admit a geometric representation that directly yields the maximal univalent domain.

Editorial extensions

If this is right

The radius of convergence of the Maclaurin series equals the distance from the origin to the boundary of the maximal univalent domain.
The functions commute with rotation by the angle determined by the geometric construction.
The functions admit analytic continuation across the boundary of the univalent domain into larger regions.
Periodicity relations hold throughout the maximal domains and are inherited from the underlying geometric picture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same geometric construction may supply maximal domains for other families of generalized functions once a comparable representation is found.
When the two parameters reduce to the classical sine or cosine case, the domains and periodicity statements should recover the familiar real-line behavior as a special case.
Knowledge of the precise boundary could be used to locate the nearest singularities and thereby improve numerical continuation algorithms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main takeaway is that Ding presents a unified geometric approach to analytically continue generalized trigonometric functions with two parameters to their largest univalent domains in the complex plane, then extracts consequences on Maclaurin radius, commutation with rotation, continuation past univalence, and periodicity. The abstract frames the construction itself as the new piece rather than a restatement of prior results. If the geometry works as described, it supplies a direct method for this specific family that could be cleaner than series-based or case-by-case extensions. The listed consequences follow routinely once the domains are fixed, so the value sits in the construction. The abstract gives no equations or lemmas, which makes it impossible to verify whether the geometric representation actually produces the claimed maximal domains without extra assumptions or steps. That is the main soft spot, but it is not a load-bearing flaw visible from the given material; the stress-test note correctly flags no internal inconsistency or circularity in the abstract. The work stays within standard complex analysis territory and does not invent entities or fit parameters to force the outcome. This is for complex analysts already working on generalizations of trigonometric functions or univalent mappings. A reader in that niche could adapt the geometric technique if the details hold up. I would bring it to a reading group focused on complex analysis or special functions for discussion of the method. I would not cite it in my own work in the next year unless the full proofs turn out to be especially clean or reusable. It deserves peer review because the claim is concrete and narrow enough that referees can check the geometry and derivations without needing broad new machinery.

Referee Report

0 major / 1 minor

Summary. The manuscript claims that a unified geometric approach allows analytic continuation of a class of two-parameter generalized trigonometric functions to their maximal domains on which they are univalent. Consequences are then deduced for the radius of convergence of the Maclaurin series, commutation with rotation, continuation beyond the univalent domain, and periodicity.

Significance. If the geometric construction rigorously yields the claimed maximal univalent domains, the work would supply a direct, parameter-free method for determining domains of univalence for these generalized functions, potentially unifying prior results on their analytic properties in complex analysis.

minor comments (1)

Abstract: the opening sentence refers to 'a class of generalized trigonometric functions with two parameters' without indicating their explicit form or the geometric representation; adding one sentence with the defining relation would make the claim immediately concrete for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper advances a unified geometric representation of two-parameter generalized trigonometric functions that directly determines their maximal univalent domains and standard consequences (radius of convergence, commutation, periodicity). No equations reduce a claimed prediction to a fitted input by construction, no self-citation chain supplies a load-bearing uniqueness theorem or ansatz, and no renaming of known results occurs. The approach is presented as independent of external fitted values or prior author-specific results that would collapse the derivation.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or new entities; the work is presented as a geometric construction within standard complex analysis.

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Cite this review

Pith. "Pith review of Analytic Continuation of Generalized Trigonometric Functions." pith.science (2026). https://pith.science/paper/RJHIDVXB

@misc{pith2026210615109,
  author       = {Pith},
  title        = {Pith review of: Analytic Continuation of Generalized Trigonometric Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJHIDVXB}},
  note         = {Machine review of arXiv:2106.15109}
}

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Reviewed May 25, 2026 · model on record in the stance chip above.