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首页> 外文期刊>Real analysis exchange >CONTOUR INTEGRATION UNDERLIES FUNDAMENTAL BERNOULLI NUMBER RECURRENCE
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CONTOUR INTEGRATION UNDERLIES FUNDAMENTAL BERNOULLI NUMBER RECURRENCE

机译：轮廓整合说明基本的BERNOULLI数递归

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摘要

One solution to a relatively recent American Mathematical Monthly problem [6], requesting the evaluation of a real definite integral, could be couched in terms of a contour integral which vanishes a priori. While the required real integral emerged on setting to zero the real part of the contour quadrature, the obligatory, simultaneous vanishing of the imaginary part alluded to still another pair of real integrals forming the first two entries in the infinite log-sine sequence, known in its entirety. It turns out that identical reasoning, utilizing the same contour but a slightly different analytic function thereon, sufficed not only to evaluate that sequence anew, on the basis of a vanishing real part, but also, in setting to zero its conjugate imaginary part, to recover the fundamental Bernoulli number recurrence. The even order Bernoulli numbers B_(2k) entering therein were revealed on the basis of their celebrated connection to Riemann's zeta function ζ(2k). Conversely, by permitting the related Bernoulli polynomials to participate as integrand factors, Euler's connection itself received an independent demonstration, accompanied anew by an elegant log-sine evaluation, alternative to that already given. And, while the Bernoulli recurrence is intended to enjoy here the pride of place, this note ends on a gloss wherein all the motivating real integrals are recovered yet again, and in quite elementary terms, from the Fourier series into which the Taylor development for Log(l - z) blends when its argument z is restricted to the unit circle.
机译：一个解决较新的《美国数学月刊》问题的方法[6]，要求评估一个实际的定积分，可以用先验消失的轮廓积分来表示。在将轮廓积分的实部设置为零时出现了所需的实数积分，而虚部的强制性同时消失则隐含了另一对实数积分，它们构成了无限对数正弦序列中的前两个项。完整。事实证明，利用相同轮廓但略有不同的解析函数进行相同的推理，不仅可以基于消失的实部重新评估该序列，而且还可以将其共轭虚部设为零。恢复基本的伯努利数递归。根据它们与Riemann的zeta函数ζ（2k）的著名联系，揭示了进入其中的偶数伯努利数B_（2k）。相反，通过允许相关的伯努利多项式作为整数因子参与，欧拉的连接本身得到了独立的演示，并伴有优雅的对数正弦评估，这是已经给出的替代方法。而且，虽然伯努利式的递归旨在在这里享受住所的骄傲，但本笔记以一种光泽结尾，其中所有具有激励作用的实数积分又从相当基本的意义上从傅里叶级数中恢复了出来，泰勒对数的泰勒展开式（l-z）当其参数z限制为单位圆时会混合。

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