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arxiv: 1907.02553 · v1 · pith:BQKM24DSnew · submitted 2019-07-04 · 🧮 math.HO · math.CA· math.CO

The Newton integral and the Stirling formula

Martin Klazar This is my paper

Pith reviewed 2026-05-25 02:13 UTC · model grok-4.3

classification 🧮 math.HO math.CAmath.CO
keywords Newton integralStirling formulan!asymptotic formulagamma functionFubini theoremlogarithmic sumprimitive function
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The pith

The Newton integral, defined as the difference of any primitive at the endpoints, suffices to derive the Stirling asymptotic formula for n!.

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

The paper establishes that the Newton integral is the logically simplest form of integration sufficient to prove the Stirling asymptotic formula. It reviews two derivations in full detail inside this framework: one that approximates the sum of logarithms from 1 to n by an integral, and another that uses the gamma function together with a Fubini-type interchange. A reader would care because the result isolates the minimal property of integration actually required for the classical approximation of n!, showing that stronger notions of area or measure are not needed. The paper also records two further integral representations of the factorial.

Core claim

The Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval, is the logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. The author reviews in its framework in detail two derivations of the Stirling formula. The first approximates log(1)+log(2)+...+log(n) with an integral and the second uses the classical gamma function and a Fubini-type result. Two more integral representations of n! are mentioned.

What carries the argument

The Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval.

If this is right

  • The sum of logarithms from 1 to n can be compared to an integral using solely the Newton definition.
  • The gamma-function derivation of Stirling's formula, including the Fubini interchange, proceeds inside the same minimal framework.
  • The Stirling asymptotic formula for n! follows directly from these two derivations.
  • Two additional integral representations of n! are available that fit the Newton-integral setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result indicates that the logical dependence of Stirling's formula on integration theory may be weaker than usually assumed.
  • Similar minimal integral tools might suffice for other asymptotic formulas that currently invoke Riemann or Lebesgue integrals.
  • The approach could be tested on related results such as the asymptotic expansion of the gamma function at positive integers.

Load-bearing premise

That both the sum-to-integral comparison and the gamma-function Fubini argument can be carried out completely inside the Newton-integral framework without extra analytic tools.

What would settle it

A concrete step in either reviewed derivation that cannot be justified using only the endpoint-difference definition of the integral and instead requires a property such as continuity of the integrand or dominated convergence that lies outside the Newton definition.

read the original abstract

We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval. We review in its framework in detail two derivations of the Stirling formula. The first approximates log(1)+log(2)+...+log(n) with an integral and the second uses the classical gamma function and a Fubini-type result. We mention two more integral representations of n!.

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

1 major / 1 minor

Summary. The manuscript claims that the Newton integral—defined strictly as the difference F(b)−F(a) for any primitive F—is the logically simplest integral sufficient to deduce the Stirling asymptotic formula for n!. It reviews two derivations in detail within this framework: one approximating ∑log k by an integral, and the second using the gamma-function integral representation of n! together with a Fubini-type interchange. Two further integral representations of n! are mentioned.

Significance. If the derivations can be executed using only the stated endpoint-difference definition, the paper would usefully isolate the minimal analytic content needed for Stirling’s formula and serve a pedagogical or foundational purpose. The explicit review of both derivations is a strength, but the overall significance is tempered by the expository character and the need to verify that no auxiliary convergence or interchange results are tacitly imported.

major comments (1)
  1. [Gamma function derivation] Gamma-function derivation (second reviewed proof): the Fubini-type interchange applied to the iterated integral representation of n! is invoked without explicit justification from the Newton-integral definition alone. The bare statement F(b)−F(a) does not entail the legitimacy of changing the order of integration; standard justifications (monotone convergence, uniform convergence, or Fubini for positive integrands) lie outside the given definition. This step is load-bearing for the central claim that both derivations remain strictly inside the Newton-integral framework.
minor comments (1)
  1. The abstract states that two additional integral representations of n! are mentioned; the manuscript should indicate whether these representations are also shown to be derivable from the Newton integral or are merely listed for context.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for isolating the key technical point concerning the gamma-function derivation. We address this comment directly below.

read point-by-point responses

  1. Referee: Gamma-function derivation (second reviewed proof): the Fubini-type interchange applied to the iterated integral representation of n! is invoked without explicit justification from the Newton-integral definition alone. The bare statement F(b)−F(a) does not entail the legitimacy of changing the order of integration; standard justifications (monotone convergence, uniform convergence, or Fubini for positive integrands) lie outside the given definition. This step is load-bearing for the central claim that both derivations remain strictly inside the Newton-integral framework.

    Authors: We agree that the endpoint-difference definition alone does not automatically authorize reordering of integrals. The manuscript already describes the step as employing 'a Fubini-type result,' thereby signaling that an additional principle is used. In revision we will supply an explicit, self-contained verification for the concrete iterated-integral representation appearing in the gamma-function argument. Because the relevant integrands are positive and continuous, the two iterated integrals can each be evaluated by finding explicit antiderivatives and applying the Newton definition at the endpoints; direct comparison of the resulting closed-form expressions then confirms equality without appeal to general convergence theorems. This keeps the argument elementary while making the additional step fully transparent. We will also adjust the surrounding prose to state precisely which minimal extra principle is required, thereby sharpening rather than weakening the paper's claim about the Newton integral as the principal analytic tool. revision: yes

Circularity Check

0 steps flagged

No circularity: expository review of standard derivations inside the stated Newton-integral definition.

full rationale

The paper defines the Newton integral explicitly as F(b)−F(a) for any primitive F and then reviews two known derivations of Stirling's formula (sum-to-integral comparison and gamma-function representation) inside that framework. No equations are fitted to data and then re-labeled as predictions; no self-citation chain is invoked to justify a uniqueness theorem or an ansatz; the Fubini-type step is presented as part of the reviewed argument rather than smuggled in by prior self-work. The central claim is therefore an assertion of sufficiency of a minimal definition, not a reduction of the target result to its own inputs by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the paper invokes the existence of primitives (antiderivatives) and standard integral properties for the gamma function and Fubini-type interchange.

axioms (2)
  • standard math Every continuous function on an interval possesses a primitive (antiderivative).
    Required by the definition of the Newton integral given in the abstract.
  • domain assumption Fubini-type interchange is valid for the gamma-function representation.
    Mentioned as part of the second derivation.

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