There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Table Of Integrals Series And Products

Tables of integrals

DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

/pdf/the-riemann-zeta-function-3l59kypwcz.pdf

The Riemann zeta function

Introduction Permit me to begin by telling you a little about the history of the book1 on which this talk2 is based. It will help you understand why I am so delighted to be presenting this talk. On the very day thirty-five years ago when my History of Vector Analysis was published, a good friend with the very best intentions helped me put the book in perspective by innocently asking: “Who was Vector?” That question might well have been translated into another: “Why would any sane person be interested in writing such a book?” Moreover, a few months later, one of my students recounted that while standing in the corridor of the Notre Dame Library, he overheard a person expressing utter astonishment and was staring at the title of a book on display in one of the cases. The person was pointing at my book, and asking with amazement: “Who would write a book about that?” It is interesting that the person who asked “Who was Vector?” was trained in the humanities, whereas the person in the library was a graduate student in physics. My student talked to the person in the library, informing him he knew the author and that I appeared to be reasonably sane. These two events may suggest why my next book was a book on the history of ideas of extraterrestrial intelligent life. My History of Vector Analysis did not fare very well with the two people just mentioned, nor did it until now lead to any invitations to speak. The humanities departments at Notre Dame assumed that my subject was too technical, the science and math departments must have assumed that it was not technical enough. In any case, never in the thirty-five intervening years did I ever have occasion to talk on my topic. My response when recently asked to talk about the subject was partly delight—I had always wanted to do this—but also some hesitation—this was a topic I researched nearly forty years ago! But it has turned out to be fun. Publishing the book has also proved interesting. Although it is not for everyone, the hardbound printing of about 1200 copies gradually nearly sold out, based partly on a number of very favorable reviews. It is rare that academic books sell that many copies. As it was about to go out of print, I hit on the idea of asking Dover whether they would want to take it over. This resulted in its re-publication in 1985 with a new preface updating the bibliography; by that time, there had appeared a few dozen papers and books shedding new light on various aspects of the subject. In the early 1990s, a curious development occurred. Nearly twenty-five years after the book had been published, a research center in Paris (La Maison des Sciences de l’Homme) announced a prize competition for a study on the history of complex and hypercomplex numbers). As you can imagine, I was quite pleased to submit my book. Some months later I was notified that I was being awarded a Jean Scott Prize, which included a check for $4000. At this point, Dover decided to do a new printing of the book, which includes an announcement of the prize. In any case, the book has now been continuously in print for 35 years and has led to all sorts of interesting letters and exchanges.

A History of Vector Analysis. By M. J. Crowe. Pp. xvii, 270. 123s. 1967 (University of Notre Dame Press).

In this article we present a method for proving a class of inequalities of the form (1). The method is based on the precise approximations of the sine and cosine functions by Maclaurin polynomials of given order. By using this method we present new proofs of some inequalities of C.-P. Chen, W.-S. Cheung [J. Inequal. Appl. 2012:72 (2012)] and Z.-J. Sun, L. Zhu [ISRN Math. Anal. (2011)].

/pdf/a-method-for-proving-some-inequalities-on-mixed-20yfokwulk.pdf

A method for proving some inequalities on mixed trigonometric polynomial functions

In this paper we propose a new method for sharpening and refinements of some trigonometric inequalities We apply these ideas to some inequalities of Wilker–Cusa–Huygens type

/pdf/extensions-of-the-natural-approach-to-refinements-and-462urhqs9d.pdf

Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities.

The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities of the form $$\sum_{i=1}^{n}\alpha _{i}x^{p_{i}} \cos ^{q_{i}} x\sin ^{r_{i}} x>0 $$
 by reducing them to polynomial inequalities. Finally, we show the great applicability of this algorithm and, as an example, we use it to analyze some new rational (Pade) approximations of the function cos2
 x and to improve a class of inequalities by Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.

/pdf/the-natural-algorithmic-approach-of-mixed-trigonometric-ikqrpt02me.pdf

The natural algorithmic approach of mixed trigonometric-polynomial problems.

In this article we consider mathematical fundamentals of one method for proving inequalities by computer, based on the Remez algorithm. Using the well-known results of undecidability of the existence of zeros of real elementary functions, we demonstrate that the considered method generally in practice becomes one heuristic for the verification of inequalities. We give some improvements of the inequalities considered in the theorems for which the existing proofs have been based on the numerical verifications of Remez algorithm.

Some Notes on a Method for Proving Inequalities by Computer

In this paper, we give some sharper refinements and generalizations of inequalities related to Shafer-Fink’s inequality for the inverse sine function stated in Theorems 1, 2, and 3 of Bercu (Math. Probl. Eng. 2017: Article ID 9237932, 2017).

/pdf/refinements-and-generalizations-of-some-inequalities-of-2mcp6h3lcm.pdf

Branko Malesevic

Papers

A method for proving some inequalities on mixed trigonometric polynomial functions

Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities.

The natural algorithmic approach of mixed trigonometric-polynomial problems.

Some Notes on a Method for Proving Inequalities by Computer

Refinements and generalizations of some inequalities of Shafer-Fink's type for the inverse sine function.