On the generalized inverse tangent integral and Catalan's constant

Technical Summary: On the generalized inverse tangent integral and Catalan's constant

Problem Statement
The paper addresses the analytical structure of the inverse tangent integral, defined as $Ti_2(y) = \int_0^y \frac{\arctan(x)}{x} dx$, and its special value at $y=1$, which yields Catalan's constant $G$. While the connection between $Ti_2$ and the imaginary part of the dilogarithm is well-established, the author seeks to develop new identities that bridge arctangent-type integrals with polylogarithmic structures, Fourier-type expansions, and Hurwitz zeta functions. The primary goal is to establish a systematic framework for decomposing generalized inverse tangent integrals into explicit formulas involving elementary terms, dilogarithms, and infinite series of correction terms.

Methodology
The paper employs a generating-function perspective combined with complex analysis and classical special function theory. The methodology is divided into two main approaches:

1. Auxiliary Integral with Tunable Endpoint: The author introduces a two-parameter auxiliary integral $I(a, b) = \int_0^b \arctan\left(\frac{a + \cos \beta}{\sin \beta}\right) d\beta$. By analyzing the partial derivatives of this integral with respect to the parameter $a$ and comparing them to the derivatives of a function constructed from the real part of a dilogarithm, the author establishes an identity linking $I(a, b)$ to $Li_2$. This allows for the representation of $Ti_2(a)$ through a "tunable" endpoint $b(a)$, determined by an admissibility condition involving the imaginary part of $Li_2(1+ia)$.

2. Spectral/Mode Expansion via Euler's Product: The second approach utilizes Euler's infinite product formula for the sine function to derive a partial fraction expansion for the cotangent function. By applying this to the argument of the inverse tangent integral, the author decomposes the integral into a principal hyperbolic term and an infinite series of correction terms indexed by the poles of the cotangent. This method relies on termwise differentiation and integration of complex arguments involving $\cot(\alpha + ix)$.

Key Contributions and Results

• Theorem 1 (Inverse Tangent Integral with Tunable Endpoint): The paper establishes a representation for $Ti_2(a)$ for $a > 0$ in terms of an auxiliary integral $I(a, b)$ and elementary logarithmic terms. Specifically, it proves that for an admissible set of $a$, there exists a unique $b \in (0, \pi)$ such that:
  $$Ti_2(a) = \arctan(a)\ln(a) + I(a, b) - \frac{\pi b}{2} + \frac{b^2}{2} - \frac{\pi}{4}\ln(a^2+1)$$
  This identity balances the inverse tangent integral against real and imaginary parts of dilogarithmic combinations.

• Corollary 1 (Evaluation of Catalan's Constant): By specializing Theorem 1 to $a=1$, the author derives a specific evaluation for Catalan's constant $G$. The result expresses $G$ in terms of a unique parameter $b(1)$ derived from the auxiliary integral:
  $$G = \frac{b(1)^2}{4} - \frac{\pi}{4}\ln(2)$$
  where $b(1) = \sqrt{4G + \pi \ln(2)}$.

• Theorem 2 (Generalized Inverse Tangent Integral): The paper presents a decomposition formula for the generalized integral $\int_0^A \frac{1}{x}\arctan(x/\alpha) dx$. This integral is expressed as the sum of a hyperbolic inverse tangent contribution and an infinite series of standard inverse tangent integrals:
  $$Ti_2\left(\frac{A}{\alpha}\right) = \int_0^A \frac{1}{x}\arctan(\cot(\alpha)\tanh(x)) dx + \sum_{k=1}^\infty \left[ Ti_2\left(\frac{A}{k\pi - \alpha}\right) - Ti_2\left(\frac{A}{k\pi + \alpha}\right) \right]$$

• Corollary 3 (Catalan's Family): By setting $A = \alpha = \pi/n$, the author generates a family of identities for Catalan's constant. Notably, for $n=2$, the principal hyperbolic term vanishes, yielding a pure series representation:
  $$G = \sum_{k=1}^\infty \left[ Ti_2\left(\frac{1}{2k-1}\right) - Ti_2\left(\frac{1}{2k+1}\right) \right]$$

• Corollary 4 (Clausen's Reduction): The paper provides a reduction of $Ti_2(\tan \theta)$ into a form involving Clausen's function $Cl_2$, linking the inverse tangent integral directly to the Clausen integral:
  $$Ti_2(\tan \theta) = \theta \ln(\tan \theta) + \frac{1}{2}Cl_2(2\theta) + \frac{1}{2}Cl_2(\pi - 2\theta)$$

• Lemma 1 (Representation of Catalan's Constant): The paper derives a complex representation of $G$ involving the exponential integral $Ei$, the Gamma function $\Gamma$, and the Hurwitz zeta function $\zeta(s, c)$. This representation arises from expanding the series terms of Theorem 2 into power series and utilizing Kummer's Fourier series for the log-gamma function.

Significance
The paper claims that its primary significance lies not merely in the isolation of new formulas, but in the demonstration of a "coherent analytic pathway" connecting elementary trigonometric identities to deeper expansions in special functions. The work places Catalan's constant within a broader realm of identities governed by dilogarithmic and zeta-functional phenomena. The methodology, particularly the use of a generating-function mechanism to transition from arctangent integrals to dilogarithmic representations, is presented as a structural bridge analogous to orthogonal-polynomial expansions in special-function theory. The results offer systematic decompositions of integral quantities into principal terms and structured infinite series, providing new perspectives on the analytic properties of $Ti_2$ and $G$.