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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Closed-form summation of some trigonometric series
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by Djurdje Cvijović and Jacek Klinowski PDF
Math. Comp. 64 (1995), 205-210 Request permission

Abstract:

The problem of numerical evaluation of the classical trigonometric series \[ {S_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\sin (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}}\quad {\text {and}}\quad } {C_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},} \] where $\nu > 1$ in the case of ${S_{2n}}(\alpha )$ and ${C_{2n + 1}}(\alpha )$ with $n = 1,2,3, \ldots$ has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when $\alpha$ is equal to a rational multiple of $2\pi$, these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving ${C_\nu }(\alpha )$ and ${S_\nu }(\alpha )$ in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 205-210
  • MSC: Primary 65B10; Secondary 33E20, 65D20
  • DOI: https://doi.org/10.1090/S0025-5718-1995-1270616-8
  • MathSciNet review: 1270616