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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.		Different types do no salt do saltless water softeners really work of salt. The hard water" that you will not require homes to call the specialist services of Plumber Friendswood.<br><br>My website - [http://theblogbuddy.com/basic-ideas-for-prudent-secrets-in-ministry Pathways To Wholeness Ministry]

	~~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]].~~

	~~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~~.
	~~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>~~

	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]].

	~~==References==~~
	* Daniel Alpay (editor) (2006) ''Wavelets, Multiscale systems and Hypercomplex Analysis'', Springer, ISBN 9783764375881 .
	* 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).
	* 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]].
	* Sorin D. Gal (2004) ''Introduction to the Geometric Function theory of Hypercomplex variables'', Nova Science Publishers, ISBN 1-59033-398-5.
	* 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.
	* Birkhauser Mathematics (2011) ''Hypercomplex Analysis and Applications'', series with editors Irene Sabadini and Franciscus Sommen.
	* Irene Sabadini & Michael V. Shapiro & F. Sommen (editors) (2009) ''Hypercomplex Analysis'', Birkhauser ISBN 978-3-7643-9892-7.
	* Springer (2012) ''Advances in Hypercomplex Analysis'', eds Sabadini, Sommen, Struppa.

	~~==External links==~~
	* 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".~~
	* Roman Lavika (2011) [http://www.karlin.mff.cuni.cz/~lavicka/publikace/habilitation1-~~54.pdf Hypercomplex Analysis: Selected Topics] ([[Habilitation]] Thesis) [[Charles University in Prague]].~~


	~~[[Category:Functions and mappings]~~]

en>Brad7777: /* References */ removed parent category

2011-11-11T22:33:41Z

References: removed parent category

New page

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.

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]].

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.
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>

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]].

==References==
* Daniel Alpay (editor) (2006) ''Wavelets, Multiscale systems and Hypercomplex Analysis'', Springer, ISBN 9783764375881 .
* 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).
* 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]].
* Sorin D. Gal (2004) ''Introduction to the Geometric Function theory of Hypercomplex variables'', Nova Science Publishers, ISBN 1-59033-398-5.
* 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.
* Birkhauser Mathematics (2011) ''Hypercomplex Analysis and Applications'', series with editors Irene Sabadini and Franciscus Sommen.
* Irene Sabadini & Michael V. Shapiro & F. Sommen (editors) (2009) ''Hypercomplex Analysis'', Birkhauser ISBN 978-3-7643-9892-7.
* Springer (2012) ''Advances in Hypercomplex Analysis'', eds Sabadini, Sommen, Struppa.

==External links==
* 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".
* Roman Lavika (2011) [http://www.karlin.mff.cuni.cz/~lavicka/publikace/habilitation1-54.pdf Hypercomplex Analysis: Selected Topics] ([[Habilitation]] Thesis) [[Charles University in Prague]].

[[Category:Functions and mappings]]

Barnes zeta function - Revision history

en>Rjwilmsi: /* References */Added 2 dois to journal cites using AWB (10094)

en>Brad7777: /* References */ removed parent category