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Mathematics - Analysis | Hypernumbers and Extrafunctions - Extending the Classical Calculus

Hypernumbers and Extrafunctions

Extending the Classical Calculus

Burgin, Mark

2012, VII, 160 p.

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  • Designed to introduce the reader to hypernumbers and extrafunctions, which is another rigorous mathematical approach to operations with infinite values
  • Shows that even in the most standard case of real analysis, hypernumbers and extrafunctions significantly extend the scope and increase the power not only of the classical calculus but of its moderngeneralizations and extensions, such as distribution theory or gauge integration
  • Used for enhancing traditional courses of calculus for undergraduates, as well as for teaching a separate course for graduate students

“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.

Content Level » Graduate

Keywords » differentiation - extrafuntion - hypernumber - integration - topology

Related subjects » Analysis - Dynamical Systems & Differential Equations - Theoretical, Mathematical & Computational Physics

Table of contents 

-1. Introduction: How mathematicians solve ”unsolvable” problems.-2.  Hypernumbers(Definitions and typology,Algebraic properties,Topological properties).-3. Extrafunctions(Definitions and typology, Algebraic properties, Topological properties).-4.  How to differentiate any real function (Approximations, Hyperdifferentiation).-5. How to integrate any continuous real function (Partitions and covers, Hyperintegration over finite intervals, Hyperintegration over infinite intervals). -6. Conclusion: New opportunities.- Appendix.- References.

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