Classical Analytic Methods Of Finite Trigonometric Sums

By means of classical analytic methods, such as Cauchy residue theorem, partial fraction, formal power series and basic hypergeometric series, this dissertation investigates theproblems on combinatorial computations of trigonometric identities with free parameters,closed formulae of finite trigonometric sums as well as some other kinds of trigonometricsum identities. The content in detail is as follows:The first chapter contains a simple survey on the history of trigonometric identities,as well as notations, terminology and formulae preparatory to be used later.In the second chapter, by means of Cauchy residue theorem, author devises contour integration and integrand function to establish two trigonometric identities involvingdouble free parameters, which yields a series of trigonometric sum formulae. Author con-sequently establishes closed formulae of finite trigonometric Sums of odd order, which canbe expressed as multiple convolution of binomial coefficients with an extra free parameter.In 1999, Chu and Marini derived numerous important formulae for evaluating trigono-metric sums through partial fraction decompositions. As continuation and extension ofthis approach, in the third chapter, author develops parametric decompositions of partialfractions, which lead to several trigonometric sum formulae of even order and involvingextra free parameters.In 2002, Berndt and Yeap gave several closed formulae of finite trigonometric sumsin which the summation domain are taken as some partitions of an integer. In the fourthchapter, by means of cyclotomic polynomial method and combinatorial computation tech-nique, author deduces some closed formulae of finite trigonometric sums with double freeparameters and of order 4n.The fifth chapter contributes to further applications of partial fraction method andCauchy residue theorem to identities involving trigonometric function fraction. Authorfirst discusses the decompositions of trigonometric function fractions with irregular fac-tors, by which and Cauchy residue theorem the author obtains several trigonometric sumidentities in which the denominators are product of linear and quadratic sin-functions withirregularly distributed variables.