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| In mathematics, '''hypercomplex analysis''' is the extension of [[real analysis]] and [[complex analysis]] to the study of functions where the [[argument of a function|argument]] is a [[hypercomplex number]]. The first instance is functions of a [[quaternion variable]], where the argument is a [[quaternion]]. A second instance involves functions of a [[motor variable]] where arguments are [[split-complex number]]s.
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| In [[mathematical physics]] there are hypercomplex systems called [[Clifford algebra]]s. The study of functions with arguments from a Clifford algebra is called [[Clifford analysis]].
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| A [[matrix (mathematics)|matrix]] may be considered a hypercomplex number. For example, study of [[2 × 2 real matrices#Functions of 2 × 2 real matrices|functions of 2 × 2 real matrices]] shows that the [[topology]] of the [[space (mathematics)|space]] of hypercomplex numbers determines the function theory. Functions such as [[square root of a matrix]], [[matrix exponential]], and [[logarithm of a matrix]] are basic examples of hypercomplex analysis.
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| The function theory of [[diagonalizable matrices]] is particularly transparent since they have [[eigendecomposition]]s. Suppose <math>\textstyle T = \sum _{i=1}^N \lambda_i E_i</math> where the E<sub>i</sub> are [[projection (linear algebra)|projection]]s. Then for any [[polynomial]] <math>\textstyle f, \quad f(T) = \sum_{i=1}^N f(\lambda_i ) E_i .</math>
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| Modern terminology is ''algebra'' for "system of hypercomplex numbers", and the algebras used in applications are often [[Banach algebra]]s since [[Cauchy sequence]]s can be taken to be convergent. Then the function theory is enriched by [[sequence]]s and [[series (mathematics)|series]]. In this context the extension of holomorphic functions of a complex variable is developed as the [[holomorphic functional calculus]]. Hypercomplex analysis on Banach algebras is called [[functional analysis]].
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| ==References==
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| * Daniel Alpay (editor) (2006) ''Wavelets, Multiscale systems and Hypercomplex Analysis'', Springer, ISBN 9783764375881 .
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| * Enrique Ramirez de Arellanon (1998) ''Operator theory for complex and hypercomplex analysis'', [[American Mathematical Society]] (Conference proceedings from a meeting in Mexico City in December 1994).
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| * Geoffrey Fox (1949) ''Elementary Function Theory of a Hypercomplex Variable and the Theory of Conformal Mapping in the Hyperbolic Plane'', M.A. thesis, [[University of British Columbia]].
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| * Sorin D. Gal (2004) ''Introduction to the Geometric Function theory of Hypercomplex variables'', Nova Science Publishers, ISBN 1-59033-398-5.
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| * R. Lavika & A.G. O’ Farrell & I. Short(2007) "Reversible maps in the group of quaternionic Möbius transformations", [[Mathematical Proceedings of the Cambridge Philosophical Society]] 143:57–69.
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| * Birkhauser Mathematics (2011) ''Hypercomplex Analysis and Applications'', series with editors Irene Sabadini and Franciscus Sommen.
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| * Irene Sabadini & Michael V. Shapiro & F. Sommen (editors) (2009) ''Hypercomplex Analysis'', Birkhauser ISBN 978-3-7643-9892-7.
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| * Springer (2012) ''Advances in Hypercomplex Analysis'', eds Sabadini, Sommen, Struppa.
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| ==External links==
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| * Chapman University [http://www.chapman.edu/scst/centers-of-excellence/cecha/index.aspx Center of Excellence in Hypercomplex Analysis], includes Daniele Struppa, Chancellor of [[Chapman University]], Chapman faculty, and several "external faculty".
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| * Roman Lavika (2011) [http://www.karlin.mff.cuni.cz/~lavicka/publikace/habilitation1-54.pdf Hypercomplex Analysis: Selected Topics] ([[Habilitation]] Thesis) [[Charles University in Prague]].
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| [[Category:Functions and mappings]]
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