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Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals

This paper presents compact, non-recursive coefficient formulae for hyperbolic and logarithmic tangent integrals by expressing their values as linear combinations of odd zeta and even Dirichlet beta values, utilizing Chebyshev-arcsine extractions and cotangent power series expansions.

Original authors: Luc Ramsès Talla Waffo

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luc Ramsès Talla Waffo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake a giant, complex cake, but the recipe is written in a secret code. Instead of listing ingredients like "2 cups of flour," the recipe says, "Take the number from the 4th row of a hidden table, multiply it by the result of a nested sum from the 3rd column, and then subtract the value of a recursive loop." It's a nightmare. You can't see what the cake will taste like, and if you want to change the size of the cake, you have to recalculate the entire secret code from scratch.

This is exactly the problem mathematicians have been facing with a specific family of integrals (which are like fancy, infinite-area calculations) involving hyperbolic functions and logarithms. For a long time, the numbers that tell us the value of these integrals were hidden inside these messy, recursive "secret codes."

Enter Luc Ramsès Talla Waffo, a researcher from Darmstadt, Germany, who has found a way to rewrite the entire recipe.

The Magic Key: From Secret Codes to Single Numbers

The main discovery in this paper is a new, "compact" way to find the coefficients (the secret numbers) for these integrals. Instead of digging through a labyrinth of recursive steps or nested sums, Waffo shows that you can find the answer by looking at a single coefficient in a much simpler, explicit expression.

Think of it like this: previously, to find a specific ingredient, you had to climb a ladder, open a box, find a map, and follow a trail of breadcrumbs. Now, Waffo says, "Just look at the label on the jar."

The paper proves that for a wide range of these integrals, the messy coefficients can be replaced by a single "extraction" from a known mathematical object. Specifically, they use a combination of Chebyshev polynomials (which are like special, wavy shapes used in engineering) and arcsine functions (the inverse of sine, related to angles).

The Three Families of Integrals

Waffo organizes these integrals into three main groups, each getting a simpler "recipe":

The Shifted Hyperbolic Integrals: These involve terms like $\sinh((2k+1)x)$ divided by powers of $\cosh(x)$ .
- The Old Way: You had to calculate a long list of numbers based on previous steps.
- The New Way: The answer is simply the coefficient of a specific power of $x$ in the product of a Chebyshev polynomial and a power of $\arcsin(x)$ . It's like finding the number of red marbles in a single, well-defined jar instead of counting them from a pile of mixed marbles.
The Logarithmic Tangent Integrals: These involve integrals from 0 to $\pi/4$ with $\ln(\tan x)$ in the denominator.
- The Old Way: The coefficients were buried in complex sums.
- The New Way: Waffo shows these are also just single coefficients from a product involving Chebyshev polynomials (specifically $U_{2n-1}(1/x)$ ) and powers of $\arcsin(x)$ . This method even reveals a surprising "parity-free" rule: the very last number in the sequence is always a simple power of 2 ( $2^{r-1}$ ), regardless of whether the numbers involved are odd or even. This was a hidden pattern that the old, messy formulas obscured.
The General Tangent Power Integrals: This is the big one. It covers integrals of the form $\int_0^\infty \frac{\tanh^{m+1} x}{x^{n+1}} dx$ .
- The Old Way: These were often described using "umbral notation" (a fancy shorthand that hides the actual numbers) or required recursive calculations.
- The New Way: Waffo proves a direct formula. The integral equals a sum where the coefficients are extracted from the expansion of $(u \cot u)^{m+1}$ .
- The "Vanishing" Trick: One of the coolest features of this new formula is that it immediately explains why certain terms disappear. If you look at the formula, you can see that for small values of $p$ , the coefficients are zero. This means the "first few" zeta values (special numbers like $\zeta(3)$ , $\zeta(5)$ ) simply don't appear in the answer for certain integrals. The formula makes this disappearance obvious, whereas the old methods hid it.

What This Proves (and What It Doesn't)

It is important to note that this paper is a proof, not a simulation or a guess. The author uses rigorous mathematical tools—contour integration, residue calculus, and properties of polynomials—to demonstrate that these new formulas are exactly correct.

What is ruled out: The paper argues against the idea that these coefficients must be calculated via recursive arrays or nested sums. It proves that a single, explicit algebraic operation is sufficient and superior.
What is confirmed: The paper confirms that the values of these integrals are finite linear combinations of odd zeta values (like $\zeta(3), \zeta(5)$ ) and even Dirichlet beta values.
The "Diagonal" Case: In the specific case where the powers are equal (the "diagonal" case), the formula gives a direct, non-recursive way to calculate $\int_0^\infty (\frac{\tanh x}{x})^N dx$ , replacing the nested coefficients found in earlier work by Kyrion.

The "Why It Matters" (Without the Hype)

The paper doesn't claim to solve a physics problem or cure a disease. Instead, it solves a problem of clarity and efficiency.

By replacing hidden, recursive calculations with a single coefficient extraction, the paper makes the structure of these integrals visible. It allows mathematicians to:

Compute the numbers directly without building up from previous steps.
See exactly how the answer depends on the parameters (like $m$ and $n$ ).
Spot patterns, like the vanishing of initial terms or the simple form of the final term, which were previously invisible behind "hidden cancellations" in the old formulas.

In short, Waffo has taken a tangled ball of yarn (the recursive sums) and pulled out a single, straight thread (the explicit coefficient formula) that leads directly to the answer. It's a cleaner, faster, and more transparent way to do the math, proving that sometimes the most complex problems just need a better way of looking at them.

Technical Summary: Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals

Problem Statement
The paper addresses the evaluation of definite integrals involving hyperbolic and logarithmic tangent functions, specifically those whose values can be expressed as finite linear combinations of odd Riemann zeta values ( $\zeta(2p+1)$ ) and even Dirichlet beta values ( $\beta(2p)$ ). A persistent difficulty in this domain, as noted in prior work by Kyrion [1] and related contour evaluations [2], is that while the special values themselves are known, the accompanying coefficients are often defined by recursive arrays, nested finite sums, or auxiliary structures. These definitions obscure the computation, hide the dependence on parameters, and conceal structural features such as vanishing ranges or simple forms of initial and terminal coefficients.

Methodology
The author develops a unified framework based on coefficient extraction from explicit elementary expressions. The methodology proceeds through three main stages:

Chebyshev–Arcsine Extractions: For shifted hyperbolic integrals of the form $\int_0^\infty \frac{\sinh((2k+1)x)}{x \cosh^n x} dx$ , the paper converts coefficient sums derived from contour evaluations into single coefficient extractions involving Chebyshev polynomials of the second kind ( $U_m$ ) and powers of $\arcsin x$ . This is achieved by transforming Cauchy integral representations via the substitution $x = \sin z$ .
Laurent Coefficient Formulas: For logarithmic tangent integrals $\int_0^{\pi/4} \frac{\sin(4nx)}{\ln(\tan x)} dx$ and $\int_0^{\pi/4} \frac{\cos((4n-2)x)}{\ln(\tan x)} dx$ , the author combines multiple-angle expansions with binomial identities for Chebyshev polynomials. This yields compact Laurent coefficient formulas involving $U_{2n-1}(1/x)$ or $U_{2n-2}(1/x)$ multiplied by powers of $\arcsin x$ .
Cotangent Collapse via Integration by Parts: For the general family $\int_0^\infty \frac{\tanh^{m+1} x}{x^{n+1}} dx$ (where $m \ge n \ge 1$ and $m+n$ is even), the paper employs repeated integration by parts. This reduces the integral to a sum of shifted hyperbolic integrals. A key step involves an auxiliary polynomial $\Phi_{m,n}$ derived from the operator $L = (1-y^2)\frac{d}{dy}$ . Through a change of variables $v = \cot u$ , the remaining sum is "collapsed" into a single coefficient of the function $(u \cot u)^{m+1}$ .

Key Contributions and Results

Unified Coefficient Formulae: The primary result is the replacement of recursive or nested definitions with explicit, non-recursive coefficient extractions.
- For shifted hyperbolic integrals, coefficients are given by $[x^{2n}] U_{2k}(x)(\arcsin x)^{2p}$ and $[x^{2n-1}] U_{2k}(x)(\arcsin x)^{2p-1}$ .
- For logarithmic tangent integrals, coefficients are extracted from Laurent products involving $U_{2n-1}(1/x)$ or $U_{2n-2}(1/x)$ .
- For the general tangent power integral, the formula is:
  $\int_0^\infty \frac{\tanh^{m+1} x}{x^{n+1}} dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2 \rceil}^{(m+n)/2} \binom{2p}{n} \frac{(2^{2p+1}-1)\zeta(2p+1)}{\pi^{2p}} [u^{m+n-2p}](u \cot u)^{m+1}$
Structural Insights: The new formulae make structural properties immediately visible:
- Vanishing Ranges: The lower summation bound $p = \lceil n/2 \rceil$ directly explains the absence of initial zeta values (e.g., $\zeta(3)$ disappears when $n \ge 3$ ).
- Terminal Coefficients: The paper derives a parity-free identity for terminal coefficients: $[x] U_{r-1}(1/x)(\arcsin x)^r = 2^{r-1}$ . This unifies the terminal coefficients of both sine and cosine logarithmic tangent families.
- Initial Coefficients: For even $n$ , the coefficient of the first zeta term is shown to be independent of $m$ , simplifying to $(-1)^{n/2}(2^{n+1}-1)\zeta(n+1)/\pi^n$ .
Direct Proofs: The methodology provides direct proofs for specific coefficients (such as the coefficient of $\beta(2)/\pi$ ) that were previously established via other means, now derived solely from the coefficient-extraction representation.

Significance
The paper claims that the principal advantage of these formulae is the replacement of "recursively generated numbers" with "one local algebraic operation on a known function." This shift offers three specific benefits:

Computability: Coefficients can be computed directly from standard power series without maintaining recursive arrays.
Visibility: The dependence on all parameters ( $m, n, k$ ) remains explicit.
Structural Clarity: Phenomena difficult to detect in nested sums—such as the precise support of an expansion, the vanishing of initial terms, and the simple form of boundary coefficients—are exposed naturally.

In the diagonal case ( $m=n=N-1$ ), the results provide a direct, non-recursive form for $\int_0^\infty (\tanh x/x)^N dx$ , replacing the nested coefficients found in earlier descriptions by Kyrion [1]. The work does not propose new applications or experimental directions but focuses on reorganizing existing mathematical knowledge into a more transparent and computationally efficient algebraic form.

The Magic Key: From Secret Codes to Single Numbers

The Three Families of Integrals

What This Proves (and What It Doesn't)

The "Why It Matters" (Without the Hype)

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