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2012 | Book

Introduction: How Mathematicians Solve “Unsolvable” Problems Read chapter Read first chapter

Hypernumbers and Extrafunctions

Extending the Classical Calculus

Author: Mark Burgin

Publisher: Springer New York

Book Series : SpringerBriefs in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

“Hypernumbers and Extrafunctions” presents a rigorous mathematical approach to operate with infinite values. First, concepts of real and complex numbers are expanded to include a new universe of numbers called hypernumbers which includes infinite quantities. This brief extends classical calculus based on real functions by introducing extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics.

This book introduces a new theory that includes the theory of distributions as a subtheory, providing more powerful tools for mathematics and its applications. Specifically, it makes it possible to solve PDE for which it is proved that they do not have solutions in distributions. Also illustrated in this text is how this new theory allows the differentiation and integration of any real function. This text can be used for enhancing traditional courses of calculus for undergraduates, as well as for teaching a separate course for graduate students.

Table of Contents

Frontmatter
Chapter 1. Introduction: How Mathematicians Solve “Unsolvable” Problems
Abstract
Does not it seem like a contradiction or a paradox to solve an unsolvable problem? Yet the most courageous and creative mathematicians and scientists are the ones trying to make sense not only of vague ideas but also of paradoxes and contradictions. So let us look how they do this, using as an example the history of the number system development. Analyzing the evolution of numbers, we can see that it has been a process of deficiency elimination. Each step of this process caused absence of understanding, an open opposition, and even hostility to new numbers.
Mark Burgin
Chapter 2. Hypernumbers
Abstract
In this chapter we introduce real hypernumbers and study their properties in Sect. 2.1. Algebraic properties are explored in Sect. 2.2, and topological properties are investigated in Sect. 2.3. In a similar way, it is possible to build complex hypernumbers and study their properties (Burgin 2002, 2004, 2010).
Mark Burgin
Chapter 3. Extrafunctions
Abstract
In this chapter, we define and study different types of extrafunctions. It would be natural to speak of hyperfunctions instead of extrafunctions as mappings of hypernumbers. However, the term hyperfunction is already used in mathematics. So, we call mappings of hypernumbers by the name extrafunction. The main emphasis here is on general extrafunctions and norm-based extrafunctions, which include conventional distributions, hyperdistributions, restricted pointwise extrafunctions, and compactwise extrafunctions, which have studied before in different publications. In Sect. 3.1, the main constructions are described and their basic properties are explicated. For instance, a criterion is found (Theorem 3.1.1) for existence of an extension of the conventional functions to general extrafunctions. Relations between norm-based extrafunctions, distributions, hyperdistributions, pointwise extrafunctions, and compactwise extrafunctions are established. In Sect. 3.2, various algebraic properties of norm-based extrafunctions are obtained.
Mark Burgin
Chapter 4. How to Differentiate Any Real Function
Abstract
Here we explore what advantages hypernumbers and extrafunctions offer for differentiation of real functions. In Sect. 4.1, basic elements of the theory of approximations are presented. We consider approximations of two types: approximations of a point by pairs of points, which are called A-approximations and used for differentiation, and approximations of topological spaces by their subspaces, which are called B-approximations and used for integration.
Mark Burgin
Chapter 5. How to Integrate Any Real Function
Abstract
In this chapter, we study problems of integration, demonstrating the advantages that transition to hypernumbers and extrafunctions gives for this field. It is well known that it is possible to integrate only some real functions. Shenitzer and Steprāns (1994) explain that while in the eyes of some mathematicians the Lebesgue integral was the final answer to the difficulties associated with integration and there is no perfect integral, there were others who were not willing to give up the search for the perfect integral, one which would make all functions integrable. Although efforts of different mathematicians extended the scope of integrable functions, their results only gave additional evidence for impossibility of such a perfect integral.
Mark Burgin
Chapter 6. Conclusion: New Opportunities
Abstract
Many topics and results in the theory of hypernumbers and extrafunctions have been left beyond the scope of this little book as its goal is to give a succinct introduction into this rich and multilayered theory. Here we briefly describe some of these topics and results, articulating open problems and directions for further research.
Mark Burgin
Backmatter
Metadata
Title
Hypernumbers and Extrafunctions
Author
Mark Burgin
Copyright Year
2012
Publisher
Springer New York
Electronic ISBN
978-1-4419-9875-0
Print ISBN
978-1-4419-9874-3
DOI
https://doi.org/10.1007/978-1-4419-9875-0

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