Values of the derivatives of the cotangent at rational multiples of pi

摘要

By elementary arguments, we deduce closed-form expressions for the values of all derivatives of the cotangent function at rational multiples of pi. These formulae are considerably simpler than Similar ones which were found in a different manner by Kolbig. Also, we show that the values of cot((n))(pi x), n epsilon N, at x = 1/2, 1/3, 2/3, 1/4, 3/4, 1/6 and 5/6 are expressible in terms of the values of the Bernoulli polynomials alone. (C) 2008 Elsevier Ltd. All rights reserved.