More About Trigonometric Series and Integration

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \frac{\pi}{2} \right]$ may be determined by a finite or infinite sum involving the Riemann Zeta-function at odd positive integers.

Borwein integrals are one of the most popularly known phenomena in contemporary mathematics. They were found in 2001 by David Borwein and Jonathan Borwein and consist of a simple family of integrals involving the cardinal sine function ``sinc'', so that the first integrals are equal to $\pi$ until, suddenly, that pattern breaks. The classical explanation for this fact involves Fourier Analysis techniques. In this paper, we show that it is possible to derive an explanation for...

Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial differential equations and integration theory and its many and diverse applications in sciences and engineering fields makes it an attractive field of study and research. The purpose of these notes is to introduce the basic ideas and theore...

We observed that statement (2.9) in Theorem 2 remains valid if condition (2.7) is replaced by the weaker condition that $$f(t) \in L^1(-T, T) \quad {\rm for\ all} \quad T>0.\eqno(2.7')$$

The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_{n}$ have meromorphic representations for $n \in \mathbb{C}$, but might vanish or blow up when $n \in \mathbb{N}$. These ideas are i...

Transition from Fourier series to Fourier integrals is considered and error introduced by ordinary substitution of integration for summing is estimated. Ambiguity caused by transition from discrete function to continuous one is examined and conditions under which this ambiguity does not arise are suggested.

In this article we provide a motivation for studying integrals $\displaystyle \int_0^{\frac{\pi}{2}}x\cos^mx~dx$, $m\in \mathbb{N}$. We demonstrate several properties of this class of integrals and show that it is closely related to the class of Wallis integrals, $\displaystyle \int_0^{\frac{\pi}{2}}\cos^mx~dx$, $m\in \mathbb{N}$.

A family of general Master theorems for analytic integration over the real (or imaginary) axis with various reciprocal hyperbolic (trig) kernels ($\sinh and/or \cosh$) with varying arguments is developed. Several examples involving closed-form integrations which do not appear to exist in the standard tables are given in detail and general results are listed in an Appendix. As well, it is shown how to convert the special case of an infinite series involving ratios of Gamma fun...

We characterize precisely the possible rate of decay of the anti-analytic half of a trigonometric series converging to zero almost everywhere.

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions.