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＜電子ブック＞
Hypernumbers and Extrafunctions : Extending the Classical Calculus / by Mark Burgin
(SpringerBriefs in Mathematics. ISSN:21918201)

版	1st ed. 2012.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	2012
大きさ	VII, 160 p : online resource
著者標目	*Burgin, Mark author SpringerLink (Online service)
件　名	LCSH:Mathematical analysis LCSH:Functional analysis LCSH:Differential equations LCSH:Measure theory LCSH:Mathematical physics FREE:Analysis FREE:Functional Analysis FREE:Differential Equations FREE:Measure and Integration FREE:Mathematical Methods in Physics
一般注記	-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 “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 HTTP:URL=https://doi.org/10.1007/978-1-4419-9875-0

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