منابع مشابه
A Few Finite Trigonometric Sums
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interest...
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We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan’s notebooks and were systematically first studied by Berndt and Zaharescu; their proofs involved complex contour integration. We show how to prove most of Berndt–Zaharescu’s and some new identities by elementary metho...
متن کاملExplicit Evaluations and Reciprocity Theorems for Finite Trigonometric Sums
This evaluation can be found in standard tables of series, such as those of E. R. Hansen [44, p. 262, eq. (30.1.2)], L. B. W. Jolly [56, pp. 102–103, eq. (352)], and A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev [70, p. 646, eq. 6]. We do not know who first proved (1.1), but several proofs exist. If we replace the power 2 on the left side by an arbitrary positive even power, finding an e...
متن کاملOn Some Trigonometric Power Sums
In contrast to Fourier series, these finite power sums are over the angles equally dividing the upper-half plane. Moreover, these beautiful and somewhat surprising sums often arise in analysis. In this note, we extend the above results to the power sums as shown in identities (17), (19), (25), (26), (32), (33), (34), (35), and (36) and in the appendix. The method is based on the generating func...
متن کاملPositive trigonometric sums and applications
Some new positive trigonometric sums that sharpen Vietoris’s classical inequalities are presented. These sharp inequalities have remarkable applications in geometric function theory. In particular, we obtain information for the partial sums of certain analytic functions that correspond to starlike functions in the unit disk. We also survey some earlier results with additional remarks and comments.
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ژورنال
عنوان ژورنال: Mathematics
سال: 2017
ISSN: 2227-7390
DOI: 10.3390/math5010013