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001 978-1-4419-9875-0
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007 cr nn 008mamaa
008 120516s2012 xxu| s |||| 0|eng d
020 _a9781441998750
_9978-1-4419-9875-0
024 7 _a10.1007/978-1-4419-9875-0
_2doi
050 4 _aQA299.6-433
072 7 _aPBK
_2bicssc
072 7 _aMAT034000
_2bisacsh
082 0 4 _a515
_223
100 1 _aBurgin, Mark.
_eauthor.
245 1 0 _aHypernumbers and Extrafunctions
_h[electronic resource] :
_bExtending the Classical Calculus /
_cby Mark Burgin.
264 1 _aNew York, NY :
_bSpringer New York :
_bImprint: Springer,
_c2012.
300 _aVII, 160 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSpringerBriefs in Mathematics,
_x2191-8198
505 0 _a-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.
520 _a“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.
650 0 _aMathematics.
650 0 _aGlobal analysis (Mathematics).
650 0 _aFunctional analysis.
650 0 _aDifferential equations, partial.
650 0 _aMathematical physics.
650 1 4 _aMathematics.
650 2 4 _aAnalysis.
650 2 4 _aFunctional Analysis.
650 2 4 _aPartial Differential Equations.
650 2 4 _aMeasure and Integration.
650 2 4 _aMathematical Methods in Physics.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9781441998743
830 0 _aSpringerBriefs in Mathematics,
_x2191-8198
856 4 0 _uhttp://dx.doi.org/10.1007/978-1-4419-9875-0
912 _aZDB-2-SMA
999 _c100603
_d100603