Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals
Abstract
We develop compact coefficient-extraction formulae for several families of hyperbolic, logarithmic tangent, and Malmsten-type integrals whose values are finite linear combinations of odd zeta values and even Dirichlet beta values. The principal advantage of these formulae is that coefficients previously encoded by recursive arrays or nested finite sums are replaced by a single coefficient of an explicit elementary expression. This makes the coefficients easier to compute, keeps the dependence on the parameters visible, and reveals structural features---such as vanishing ranges, extremal coefficients, and sign patterns---without hidden cancellations. For shifted hyperbolic integrals with numerator sinh((2k+1)x), the coefficients are expressed through Chebyshev--arcsine extractions. The same mechanism yields Laurent coefficient formulae for logarithmic tangent integrals and leads to direct proofs of simple initial and terminal coefficients, including a parity-free terminal identity. For m,n≥1, m≥ n, and m+n even, we prove ∫0∞(tanhm+1x)/(xn+1) dx = (−1)(m−n)/2 ∑p=⌈ n/2⌉(m+n)/2 binom2pn (22p+1−1) (ζ(2p+1))/π2p [um+n−2p](ucot u)m+1. ∫0∞ (tanhm x)/(xncosh x) dx = (−1)(m−n)/2 ∑p=⌈ n/2⌉(m+n)/2 22pbinom2p−1n−1 β(2p)/(π2p−1) [um+n−2p] u/(sin u)(ucot u)m In the diagonal zeta case, the formula gives the family ∫0∞(tanh x/x)N dx in a direct, non-recursive form and makes the disappearance of the initial zeta values immediate.
Related
- From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials
- Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals
- Logarithmic Lefschetz fixed point formulae and resonant boundary indices
- Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians
- Newton-type inequalities for the class of modified hyperbolic p-convex functions via Riemann-Liouville fractional integrals with numerical justifications
BibTeX
@misc{waffo2026compact,
title = {{Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals}},
author = {Luc Ramsès Talla Waffo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.12306},
eprint = {2607.12306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12306},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals
AU - Luc Ramsès Talla Waffo
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2607.12306
UR - https://arxiv.org/abs/2607.12306
AB - We develop compact coefficient-extraction formulae for several families of hyperbolic, logarithmic tangent, and Malmsten-type integrals whose values are finite linear combinations of odd zeta values and even Dirichlet beta values. The principal advantage of these formulae is that coefficients previously encoded by recursive arrays or nested finite sums are replaced by a single coefficient of an explicit elementary expression. This makes the coefficients easier to compute, keeps the dependence on the parameters visible, and reveals structural features---such as vanishing ranges, extremal coefficients, and sign patterns---without hidden cancellations. For shifted hyperbolic integrals with numerator sinh((2k+1)x), the coefficients are expressed through Chebyshev--arcsine extractions. The same mechanism yields Laurent coefficient formulae for logarithmic tangent integrals and leads to direct proofs of simple initial and terminal coefficients, including a parity-free terminal identity. For m,n≥1, m≥ n, and m+n even, we prove ∫_0^∞(tanh^m+1x)/(x^n+1) dx = (−1)^(m−n)/2 ∑_p=⌈ n/2⌉^(m+n)/2 binom2pn (2^2p+1−1) (ζ(2p+1))/π^2p [u^m+n−2p](ucot u)^m+1. ∫_0^∞ (tanh^m x)/(x^ncosh x) dx = (−1)^(m−n)/2 ∑_p=⌈ n/2⌉^(m+n)/2 2^2pbinom2p−1n−1 β(2p)/(π^2p−1) [u^m+n−2p] u/(sin u)(ucot u)^m In the diagonal zeta case, the formula gives the family ∫_0^∞(tanh x/x)^N dx in a direct, non-recursive form and makes the disappearance of the initial zeta values immediate.
ER -