Finite trigonometric sums and class numbers
WebOct 25, 2024 · In this paper, we discuss the properties of the derivative of a special function, and propose a general approach to estimating a class of trigonometric sums based on the derivative of the special function. Then we apply the approach to three trigonometric sums and get three new estimates. Using the estimate of the first trigonometric sum, we … WebNov 1, 2024 · If we plug in α = π 2 into Equation (6.1) we will take into account the equation (2.9) then for m even (for m odd the sum is zero) our sums can be written in terms of Bernoulli numbers (which...
Finite trigonometric sums and class numbers
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Webtry each method in parallel until one succeeds. "ParallelBestQuality". try each method in parallel and return the best result. "IteratedSummation". use iterated univariate summation. Automatic. automatically selected method. "HypergeometricTermFinite". special finite hypergeometric term summation. WebFinite sums with arbitrary powers of trigonometric functions have attracted a growing interest over the past few decades. To gain an appreciation of the activity and importance of this topic, the reader is urged to consult [2, 3] and examine the huge number of references cited therein. As stated in the first of these references,
WebOct 10, 2024 · Two classes of finite trigonometric sums, each involving only $\sin$'s, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations. ... Number Theory (math.NT) MSC classes: 11L03, 33B10: Cite as: arXiv:2210.04659 [math.NT] WebFinite trigonometric sums, characters, class numbers, discrete Fourier analysis, convolution. ... Class numbers make an appearance in many of the theorems that will follow, and so one can think of them as representations of class numbers by trigonometric sums. Examining the ingredients of the left-hand sides of Theorems 1.1 and 1.2 …
WebFinite trigonometric sums, characters, class numbers, discrete Fourier analysis, convolution. ... Class numbers make an appearance in many of the theorems that will … WebIf a sum cannot be carried out explicitly by adding up a finite number of terms, Sum will attempt to find a symbolic result. ... Some trigonometric sums with exponential …
WebWe 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 first systematically studied by Berndt and Zaharescu where their proofs involved complex contour integration.
impact gold fastpitch storeWebFinite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric … impact goggles. motorcycleWebFeb 18, 2024 · The mathematics literature contains many generalized trigonometric sums which are evaluated through contour integration methods, algebraic methods or through discrete Fourier analysis methods. The purpose of this paper is to show how Ramanujan’s theory of theta functions can be efficiently employed to evaluate certain generalized … impact goggles ratingWebtrigonometry, matrices and determinants, number system, partial fractions, permutations, combinations and probability, quadratic equations, sequences and series, sets, functions and groups, trigonometric functions and graphs, trigonometric identities, trigonometric ratios of allied angles tests for college and university revision guide. impact gold coast quartzWebJul 7, 2016 · An interplay between the sum of certain series related to harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits us to find sharp inequalities bounding these trigonometric sums. In particular, this answers positively an … impact gold fall showcaseWebFall 2024 - MATH 1105 - Introduction to linear algebra, probability, and Markov chains that develops the parts of the theory most relevant for applications. ... Finite Mathematics for … impact gold crushersWebNov 1, 2024 · Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. ... Mathematics > Number Theory. arXiv:1811.00361 (math) ... Number Theory (math.NT) MSC classes: Primary 33B10, 11L03: Cite as: arXiv:1811.00361 [math.NT] (or arXiv:1811.00361v1 [math.NT] for this version) list slicing python python