伯努利多项式的一致收敛性
Uniform Convergence Behavior of the Bernoulli Polynomials
摘要：
伯努利多项式的根$B_n(z)$，当在复平面中绘制时，围绕一个特殊的H形曲线积累。Karl Dilcher在1987年证明了，在$\mathbb{C}$的紧子集上，伯努利多项式的渐近行为像正弦或余弦。本文建立了$B_n(nz)$的渐近行为，计算了伯努利多项式实根的分布，并证明了适当地重新标度后，复根位于曲线$e^{- 2π\text{Im}(z)} = 2πe |z|$或$e^{2π\text{Im}(z)}= 2πe |z|$上.
英文摘要：
The roots of Bernoulli polynomials, $B_n(z)$, when plotted in the complex plane, accumulate around a peculiar H-shaped curve. Karl Dilcher proved in 1987 that, on compact subsets of $\mathbb{C}$, the Bernoulli polynomials asymptotically behave like sine or cosine. Here we establish the asmptotic behavior of $B_n(nz)$, compute the distribution of real roots of Bernoulli polynomials and show that, properly rescaled, the complex roots lie on the curve $e^{- 2π\text{Im}(z)} = 2πe |z|$ or $e^{2π\text{Im}(z)}= 2πe |z|$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
