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008 170213s2015 gw | s |||| 0|eng d
020 _a9783319248684
_9978-3-319-24868-4
024 7 _a10.1007/978-3-319-24868-4
_2doi
035 _ato000560899
040 _aSpringer
_cSpringer
_dRU-ToGU
050 4 _aQA331-355
072 7 _aPBKD
_2bicssc
072 7 _aMAT034000
_2bisacsh
082 0 4 _a515.9
_223
100 1 _aLuna-Elizarrarás, M. Elena.
_eauthor.
_9467344
245 1 0 _aBicomplex Holomorphic Functions
_helectronic resource
_bThe Algebra, Geometry and Analysis of Bicomplex Numbers /
_cby M. Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac.
250 _a1st ed. 2015.
260 _aCham :
_bSpringer International Publishing :
_bImprint: Birkhäuser,
_c2015.
300 _aVIII, 231 p. 23 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
490 1 _aFrontiers in Mathematics,
_x1660-8046
505 0 _aIntroduction -- 1.The Bicomplex Numbers -- 2.Algebraic Structures of the Set of Bicomplex Numbers -- 3.Geometry and Trigonometric Representations of Bicomplex -- 4.Lines and curves in BC -- 5.Limits and Continuity -- 6.Elementary Bicomplex Functions -- 7.Bicomplex Derivability and Differentiability -- 8.Some properties of bicomplex holomorphic functions -- 9.Second order complex and hyperbolic differential operators -- 10.Sequences and series of bicomplex functions -- 11.Integral formulas and theorems -- Bibliography.
520 _aThe purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable. The result is an elementary yet comprehensive introduction to the algebra, geometry and analysis of bicomplex numbers. Around the middle of the nineteenth century, several mathematicians (the best known being Sir William Hamilton and Arthur Cayley) became interested in studying number systems that extended the field of complex numbers. Hamilton famously introduced the quaternions, a skew field in real-dimension four, while almost simultaneously James Cockle introduced a commutative four-dimensional real algebra, which was rediscovered in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The advantages of commutativity were accompanied by the introduction of zero divisors, something that for a while dampened interest in this subject. In recent years, due largely to the work of G.B. Price, there has been a resurgence of interest in the study of these numbers and, more importantly, in the study of functions defined on the ring of bicomplex numbers, which mimic the behavior of holomorphic functions of a complex variable. While the algebra of bicomplex numbers is a four-dimensional real algebra, it is useful to think of it as a “complexification” of the field of complex  numbers; from this perspective, the bicomplex algebra possesses the properties of a one-dimensional theory inside four real dimensions. Its rich analysis and innovative geometry provide new ideas and potential applications in relativity and quantum mechanics alike. The book will appeal to researchers in the fields of complex, hypercomplex and functional analysis, as well as undergraduate and graduate students with an interest in one- or multidimensional complex analysis.
650 0 _amathematics.
_9566183
650 0 _aFunctions of complex variables.
_9304502
650 0 _aMathematical physics.
_9296775
650 1 4 _aMathematics.
_9566184
650 2 4 _aFunctions of a Complex Variable.
_9304504
650 2 4 _aSeveral Complex Variables and Analytic Spaces.
_9304717
650 2 4 _aMathematical Applications in the Physical Sciences.
_9410541
700 1 _aShapiro, Michael.
_eauthor.
_9448675
700 1 _aStruppa, Daniele C.
_eauthor.
_9311109
700 1 _aVajiac, Adrian.
_eauthor.
_9467345
710 2 _aSpringerLink (Online service)
_9143950
773 0 _tSpringer eBooks
830 0 _aFrontiers in Mathematics,
_9330496
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-319-24868-4
912 _aZDB-2-SMA
999 _c415086