CLOSED-FORM SUMMATION OF SOME TRIGONOMETRIC SERIES

The problem of numerical evaluation of the classical trigonometric series [GRAPHICS] where v > 1 in the case of S-2n(alpha) and C-2n+1(alpha) with n = 1, 2, 3,... 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-v(alpha) and S-v(alpha) in terms of the generalized Rieinann 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.