More About Trigonometric Series and Integration

A translation of Kummer`s paper On certain definite integrals and infinite series

In this note, we study a certain class of trigonometric series which is important in many problems. An unproved statement in Zygmund's book [5] will be proved and generalized. Further discussions based on this problem will also be made here.

This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series that generalize a problem published in `The College Mathematics Journal'. Some of the results thus obtained have not been founded in standard references.

The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series.

In this note a general result is proved that can be used to evaluate exactly a class of highly oscillatory integrals.

Some divergent trigonometric integrals have appeared in standard tables for many years, listed as converging. We give a simple proof that these integrals diverge and trace their history. The original error was made when a (startlingly) famous mathematician incorrectly differentiated under the integral sign with some convergent integrals that depend on a paramater.

In this article a new upper bounds for the multiple trigonometrical integrals are found. The method of the work based on a new method of estimation for the areas of algebraic surfaces.

In the main part of the paper, on the basis of contour integration of complex meromorphic functions whose singularities lie onto an integration contour, in the first step, a concept of improper integrals absolute existence of meromorphic functions, as more general one with respect to the concept of improper integrals convergence (existence), is introduced into analysis. In the second step, in the case when a modulus of complex parameter tends to infinity, an interval of impro...

We give a transform of convergent trigonometric series into equivalent convergent series and sufficient conditions for the transformed series to converge faster than the original one.

A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation formulas, and assigns limits to certain unbounded or oscillating functions. Some problems and future lines of research are briefly discussed.