Format:
Online-Ressource (VII, 160 p, digital)
ISBN:
9781441998750
,
1280802642
,
9781280802645
Series Statement:
SpringerBriefs in Mathematics
Content:
-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.
Content:
“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.
Note:
Description based upon print version of record
,
Hypernumbers and Extrafunctions; Preface; Contents; Chapter 1: Introduction: How Mathematicians Solve ``Unsolvable´´ Problems; 1.1 The Structure of this Book; Chapter 2: Hypernumbers; 2.1 Definitions and Typology; 2.2 Algebraic Properties of Hypernumbers; 2.3 Topological Properties of Hypernumbers; Chapter 3: Extrafunctions; 3.1 Definitions and Typology; 3.1.1 General Extrafunctions; 3.1.2 Norm-Based Extrafunctions; 3.2 Algebraic Properties; 3.3 Topological Properties; Chapter 4: How to Differentiate Any Real Function; 4.1 Approximations; 4.2 Hyperdifferentiation
,
Chapter 5: How to Integrate Any Real Function5.1 Covers and Partitions; 5.2 Hyperintegration over Finite Intervals; 5.3 Hyperintegration over Infinite Intervals; Chapter 6: Conclusion: New Opportunities; Appendix; 1 General Concepts and Structures; 2 Logical Concepts and Structures; 3 Topological Concepts and Structures; 4 Algebraic Concepts and Structures; 5 Notations from the Theory of Hypernumbers and Extrafunctions; References; Index;
Additional Edition:
ISBN 9781441998743
Additional Edition:
Buchausg. u.d.T. Burgin, M. S. Hypernumbers and extrafunctions New York, Heidelberg [u.a.] : Springer, 2012 ISBN 9781441998743
Additional Edition:
ISBN 1441998748
Language:
English
Keywords:
Infinitesimalrechnung
DOI:
10.1007/978-1-4419-9875-0
URL:
Volltext
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