More About Trigonometric Series and Integration

In this short note, we provide an elementary complex analytic method for converting known real integrals into numerous strange and interesting looking real integrals.

The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation. Their convergence rates have been analyzed for typical cases including finite, semi-infinite, and infinite intervals. In addition, for verified automatic integration, more explicit error bounds that are computable have been recently given on a ...

The main object of the present paper is to give a complete result regarding the best approximation rate of certain trigonometric series in general complex valued continuous function space under a new condition which gives an essential generalization to $O$-regularly varying quasimonotonicity. An application is present in Section 3.

We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed.

In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.

First some definite integrals of W. H. L. Russell, almost all with trigonometric function integrands, are derived, and many generalized. Then a list is given in Russell-style of generalizations of integral identities of Amdeberhan and Moll. We conclude with a brief and noncomprehensive description of directions for further investigation, including the significant generalization to elliptic functions.

In our recent publications we have introduced the incomplete cosine expansion of the sinc function for efficient application in sampling [Abrarov & Quine, Appl. Math. Comput., 258 (2015) 425-435; Abrarov & Quine, J. Math. Research, 7 (2) (2015) 163-174]. Here we show that it can also be utilized as a flexible and efficient tool in mathematical analysis. In particular, an application of the incomplete cosine expansion of the sinc function leads to expansion series of the error...

A paper by Bruno Salvy and the author introduced measured multiseries and gave an algorithm to compute these for a large class of elementary functions, modulo a zero-equivalence method for constants. This gave a theoretical background for the implementation that Salvy was developing at that time. The main result of the present article is an algorithm to calculate measured multiseries for integrals of functions of the form h*sin G, where h and G belong to a Hardy field. The pr...

We present a replacement for traditional Riemann integrals in undergraduate calculus, which supplements naive precalculus and at the same time opens a way to more sophisticated theories such as Lebesgue integration.

In paper this paper it is considered the summation problem for trigonometric integrals with quadratic phase. This problem considered in the papers \cite{Chub},\cite{Chax},\cite{Jabbar} in particular cases. Our results generalized the results of that papers to multidimensional trigonometrical integrals.