发表机构
Technische Universität Darmstadt（达姆施塔特工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究给出双曲积分紧凑系数公式，先将特定移位积分系数重写，该方法也用于对数正切积分。最后证明一般公式，对角情况下以非递归余切系数形式给出积分族，使zeta展开初始消失显现。
AI 中文摘要
我们给出了几个双曲积分的紧凑系数公式，其值是奇数zeta值和偶数狄利克雷beta值的线性组合。首先，将分子为\(\sinh((2k + 1)x)\)的移位积分中出现的系数重写为单个切比雪夫 - 反正弦系数提取。对于\(k = 0\)，这些恒等式恢复了Kyrion研究的zeta型和beta型积分，但用反正弦\(x\)幂的系数代替了递归系数。同样的方法也产生了对数正切积分的紧凑系数公式。最后，对于\(m,n\geq1\)，\(m\geq n\)且\(m + n\)为偶数，我们证明了一般公式\[ \int_0^\infty\frac{\tanh^{m + 1}x}{x^{n + 1}}\,dx = (-1)^{(m - n)/2} \sum_{p=\lceil n/2\rceil}^{(m + n)/2} \binom{2p}{n} (2^{2p + 1}-1) \frac{\zeta(2p + 1)}{\pi^{2p}} [u^{m + n - 2p}](u\cot u)^{m + 1}. \]在对角情况下，这以非递归余切系数形式给出了\(\int_0^\infty(\tanh x/x)^N\,dx\)族，并使zeta展开的初始消失立即显现。
英文摘要
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 $m,n\geq1$, $m\geq n$, and $m+n$ even, we prove \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n} (2^{2p+1}-1) \frac{ζ(2p+1)}{π^{2p}} [u^{m+n-2p}](u\cot u)^{m+1}. \] \[\int_0^\infty \frac{\tanh^m x}{x^n\cosh x}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} 2^{2p}\binom{2p-1}{n-1} \frac{β(2p)}{π^{2p-1}} [u^{m+n-2p}] \frac{u}{\sin u}(u\cot u)^m \] In the diagonal zeta case, the formula gives the family $\int_0^\infty(\tanh x/x)^N\,dx$ in a direct, non-recursive form and makes the disappearance of the initial zeta values immediate.
Comments26 pages
