Hypernumbers and Extrafunctions: Extending the Classical Calculus

ISBN-13: 9781441998743 / Angielski / Miękka / 2012 / 160 str.

Mark Burgin

Opis

"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.

Recenzje

From the reviews:

"Burgin (UCLA) provides an introduction to the theory of hypernumbers in this short book, part of the 'SpringerBriefs in Mathematics' series. The (real) hypernumbers are an extension of the real numbers that permits operations with infinite quantities; a real general extrafunction is a partial map from the set of real hypernumbers to itself. ... Summing Up: Recommended. Advanced upper-division undergraduates, graduate students, and researchers/faculty." (D. Robbins, Choice, Vol. 50 (3), November, 2012)

Spis treści

-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.