arxiv: 2605.01830 · v1 · submitted 2026-05-03 · 🧮 math.NT

Recognition: unknown

On the generalized inverse tangent integral and Catalan's constant

Petr Vlachopulos

Pith reviewed 2026-05-09 16:24 UTC · model grok-4.3

classification 🧮 math.NT

keywords inverse tangent integralCatalan's constantdilogarithmpolylogarithmsintegral identitiesgenerating functionsarctangent integrals

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The pith

The inverse tangent integral receives explicit dilogarithmic expressions that connect it to Catalan's constant through new identities derived from auxiliary integrals.

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

The paper develops new identities for the inverse tangent integral by linking it to dilogarithmic structure. It introduces a two-parameter auxiliary arctangent integral and analyzes it through a generating-function perspective to convert it into an explicit formula involving the real part of a dilogarithm plus a companion term. At the same time, the inverse tangent integral itself is rewritten as the imaginary part of a dilogarithm at a complex argument via standard transformations. A sympathetic reader would care because these steps create a unified polylogarithmic description that systematically generates further integral representations and decompositions tied to Catalan's constant.

Core claim

By introducing an auxiliary integral depending on two parameters and analyzing it via a generating-function perspective, the inverse tangent integral is rewritten through standard integral transformations into a form governed by the imaginary part of a dilogarithm evaluated at a complex argument, while the auxiliary integral yields a compact representation in terms of the real part of a dilogarithmic expression plus a companion dilogarithm contribution. This establishes a coherent bridge between arctangent-type integrals, identities connected to Catalan's constant, and the systematic generation of new integral representations and decompositions.

What carries the argument

The two-parameter auxiliary arctangent integral, analyzed via generating functions to produce explicit dilogarithmic expressions for both the auxiliary and the inverse tangent integrals.

If this is right

The inverse tangent integral admits a clean polylogarithmic description governed by the imaginary part of a dilogarithm.
New integral representations and decompositions become available for quantities related to Catalan's constant.
Arctangent-type integrals gain a systematic method for generating further identities through the auxiliary construction.
The approach unifies several classes of special integrals under a dilogarithmic framework.

Where Pith is reading between the lines

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

The generating-function technique on the auxiliary integral could be tested on related inverse trigonometric integrals to see if similar dilog reductions appear.
The resulting expressions might simplify numerical evaluation of Catalan's constant by providing alternative integral paths.
Connections to other polylog identities at specific arguments could yield further reductions not stated in the paper.

Load-bearing premise

The auxiliary integral depending on two parameters can be converted via a generating-function perspective into the claimed explicit dilogarithmic formula without hidden restrictions on parameter domains or convergence.

What would settle it

Numerical quadrature of the auxiliary two-parameter integral at specific interior parameter values, compared against direct evaluation of the proposed real-part dilogarithm formula plus companion term, would confirm or refute the central identity.

read the original abstract

In this paper, we develop new identities for the inverse tangent integral by connecting it to the dilogarithmic (polylogarithmic) structure and to a carefully designed auxiliary arctangent integral $Ti_2(a)$ with a tunable endpoint. The core idea is based on the introduction of an auxiliary integral depending on two parameters and analyzing it via a generating-function perspective. This converts the integral into an explicit formula, yielding a compact representation in terms of the real part of a dilogarithmic expression plus a companion dilogarithm contribution. In parallel, the inverse tangent integral is rewritten through standard integral transformations into a form governed by the imaginary part of a dilogarithm evaluated at a complex argument, producing a clean polylogarithmic description. Overall, we establish a coherent bridge between arctangent-type integrals, identities connected to Catalan's constant, and the systematic generation of new integral representations and decompositions.

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.

Desk Editor's Note private letter to a colleague

The paper gives some new dilogarithmic expressions for the inverse tangent integral and ties to Catalan's constant, but the generating-function conversion step looks like it could hide domain or convergence limits.

read the letter

The main thing to know is that this paper claims new identities for the inverse tangent integral by tying it to dilogarithms and Catalan's constant. It introduces a two-parameter auxiliary integral, analyzes it with generating functions to get an explicit dilog formula, and rewrites the original integral using the imaginary part of a dilog at a complex argument. It does a decent job of creating a bridge between these arctangent-type integrals and polylog structures. The systematic generation of representations is a plus, and the compact forms it arrives at could save time for anyone computing these things numerically or symbolically. Where it might be soft is in the details of that generating-function conversion. The abstract doesn't provide much on how the parameters are handled or where the expressions converge. If the full paper skips a careful check of the radius or boundary cases, then the claimed generality could be overstated. That's the kind of thing that needs verification before the identities can be used confidently. This paper is aimed at specialists who work with special functions and polylogarithms. Someone already comfortable with dilog identities might pick up a few new tools or decompositions from it, though it won't shake up the broader field. I think it should go to peer review. The core ideas are worth having checked by experts who can test the derivations and see if the connections hold without hidden restrictions.

Referee Report

1 major / 1 minor

Summary. The paper develops new identities for the inverse tangent integral by connecting it to dilogarithmic structures via a carefully designed auxiliary two-parameter arctangent integral Ti_2(a) with tunable endpoint. It analyzes this auxiliary integral through a generating-function perspective to obtain an explicit representation as the real part of a dilogarithm plus a companion term, while rewriting the inverse tangent integral itself in terms of the imaginary part of a dilogarithm at a complex argument, thereby generating new integral representations and connections to Catalan's constant.

Significance. If the derivations hold without hidden restrictions, the work supplies a systematic polylogarithmic framework for arctangent integrals that could streamline the discovery of new identities and decompositions involving Catalan's constant. This is of moderate significance in analytic number theory and special functions, as it strengthens bridges between these areas and offers a generating-function route to explicit formulas.

major comments (1)

[Core construction and generating-function analysis] The central conversion of the two-parameter auxiliary integral to the claimed explicit dilogarithmic formula (real part plus companion term) via generating functions is load-bearing for all subsequent identities. The abstract states the integral is 'carefully designed' but provides no explicit analysis of the radius of convergence, boundary behavior at the tunable endpoint, or steps for analytic continuation; this must be supplied with precise parameter domains to justify uniform validity of the formula.

minor comments (1)

[Abstract] The abstract would be clearer if it briefly indicated the range of the two parameters for which the auxiliary integral is defined and the resulting dilogarithmic expression holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The major comment highlights the need for explicit analysis of convergence and analytic continuation for the auxiliary integral, which we address below by committing to targeted revisions that strengthen the presentation without altering the core results.

read point-by-point responses

Referee: [Core construction and generating-function analysis] The central conversion of the two-parameter auxiliary integral to the claimed explicit dilogarithmic formula (real part plus companion term) via generating functions is load-bearing for all subsequent identities. The abstract states the integral is 'carefully designed' but provides no explicit analysis of the radius of convergence, boundary behavior at the tunable endpoint, or steps for analytic continuation; this must be supplied with precise parameter domains to justify uniform validity of the formula.

Authors: We agree that the manuscript would be improved by an explicit treatment of these analytic aspects. While the derivations rely on the standard branch cuts and domains of the dilogarithm (where the generating-function approach is valid by construction), we did not include a dedicated discussion of the radius of convergence for the series expansion of the auxiliary integral, the behavior at the tunable endpoint, or the precise steps for analytic continuation. In the revised version we will insert a new subsection (immediately following the definition of the auxiliary integral) that: (i) states the parameter domains explicitly (e.g., |a| < 1 for the principal series representation, with |a| = 1 handled by Abel summation or principal-value limits); (ii) derives the radius of convergence from the generating-function differential equation; (iii) analyzes boundary behavior at the endpoint via direct estimates; and (iv) sketches the analytic continuation to other regions using the known functional equations of the dilogarithm. These additions will justify the uniform validity of the central formula and thereby support all subsequent identities and connections to Catalan's constant. revision: yes

Circularity Check

0 steps flagged

No circularity: auxiliary integral converted via generating functions yields independent dilogarithmic identities

full rationale

The paper introduces a two-parameter auxiliary arctangent integral Ti_2(a) and subjects it to a generating-function analysis that produces an explicit real-part dilogarithm formula plus companion term. This is a direct integral transformation, not a self-definition or fitted-parameter renaming. The parallel rewriting of the inverse tangent integral via imaginary-part dilogarithms at complex arguments likewise follows from standard integral manipulations. No self-citation chain, uniqueness theorem imported from the author's prior work, or ansatz smuggled via citation is invoked as load-bearing justification. The resulting bridge to Catalan's constant identities and new representations is therefore constructed from the transformed expressions rather than presupposed by them. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of dilogarithms and convergence assumptions for the auxiliary integral; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

standard math Standard analytic continuation and functional equations of the dilogarithm hold for the complex arguments appearing in the representations.
Invoked when rewriting the inverse tangent integral via imaginary parts of dilogarithms.
domain assumption The auxiliary two-parameter integral converges in the region where the generating-function analysis is performed.
Required for the conversion to an explicit dilogarithmic formula.

pith-pipeline@v0.9.0 · 5445 in / 1253 out tokens · 46252 ms · 2026-05-09T16:24:08.832257+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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