In mathematics , hypercomplex analysis is the extension of complex analysis to the hypercomplex numbers . The first instance is functions of a quaternion variable , where the argument is a quaternion (in this case, the sub-field of hypercomplex analysis is called quaternionic analysis ). A second instance involves functions of a motor variable where arguments are split-complex numbers .
In mathematical physics , there are hypercomplex systems called Clifford algebras . The study of functions with arguments from a Clifford algebra is called Clifford analysis .
A matrix may be considered a hypercomplex number. For example, the study of functions of 2 × 2 real matrices shows that the topology of the 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.[ 1] diagonalizable matrices is particularly transparent since they have eigendecompositions .[ 2]
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The modern terminology for a "system of hypercomplex numbers" is an algebra over the real numbersBanach algebras since Cauchy sequences can be taken to be convergent . Then the function theory is enriched by sequences and 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 .
See also
References
^ Felix Gantmacher (1959) The Theory of Matrices , two volumes, translator: Kurt Hirsch , Chelsea Publishing , chapter 5: functions of matrices, chapter 8: roots and logarithms of matrices^ Shaw, Ronald (1982) Linear Algebra and Group Representations , v. 1, § 2.3, Diagonalizable linear operators, pages 78–81, Academic Press ISBN 0-12-639201-3 .
Sources
Daniel Alpay (ed.) (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).
J. A. Emanuello (2015) Analysis of functions of split-complex, multi-complex, and split-quaternionic variables and their associated conformal geometries , Ph.D. Thesis, Florida State University
Sorin D. Gal (2004) Introduction to the Geometric Function theory of Hypercomplex variables , Nova Science Publishers, ISBN 1-59033-398-5 .
Lávička, Roman; O'Farrell, Anthony G. ; Short, Ian (2007). "Reversible maps in the group of quaternionic Möbius transformations" (PDF) . Mathematical Proceedings of the Cambridge Philosophical Society 143 (1): 57–69. doi :10.1017/S030500410700028X . Irene Sabadini and Franciscus Sommen (eds.) (2011) Hypercomplex Analysis and Applications , Birkhauser Mathematics.Irene Sabadini & Michael V. Shapiro & F. Sommen (editors) (2009) Hypercomplex Analysis , Birkhauser ISBN 978-3-7643-9892-7 .
Sabadini, Sommen, Struppa (eds.) (2012) Advances in Hypercomplex Analysis , Springer. 
