Hypercomplex analysis
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In mathematics, hypercomplex analysis is the basic extension of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
and complex analysis to the study of functions where the argument is a
hypercomplex number In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group represen ...
. 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 Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, there are hypercomplex systems called Clifford algebras. The study of functions with arguments from a Clifford algebra is called
Clifford analysis Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are ...
. A matrix may be considered a hypercomplex number. For example, the study of functions of 2 × 2
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
matrices shows that the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of the
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
of hypercomplex numbers determines the function theory. Functions such as
square root of a matrix In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix is said to be a square root of if the matrix product is equal to . Some authors use the name ''square root'' or the notation onl ...
,
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
, and
logarithm of a matrix In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exp ...
are basic examples of hypercomplex analysis. The function theory of diagonalizable matrices is particularly transparent since they have
eigendecomposition In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matr ...
s.Shaw, Ronald (1982) ''Linear Algebra and Group Representations'', v. 1, § 2.3, Diagonalizable linear operators, pages 78–81,
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes referen ...
.
Suppose \textstyle T = \sum _^N \lambda_i E_i where the ''E''''i'' are projections. Then for any
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
f, f(T) = \sum_^N f(\lambda_i ) E_i. The modern terminology for a "system of hypercomplex numbers" is an ''
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
over the real numbers'', and the algebras used in applications are often
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s since
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s can be taken to be convergent. Then the function theory is enriched by
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s and
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
. In this context the extension of holomorphic functions of a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
variable is developed as the
holomorphic functional calculus In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ''f'' of a complex argument ''z'' and an operator ''T'', the aim is to construct an operator, ''f''(' ...
. Hypercomplex analysis on Banach algebras is called
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
.


See also

* Giovanni Battista Rizza


References


Sources

* Daniel Alpay (ed.) (2006) ''Wavelets, Multiscale systems and Hypercomplex Analysis'', Springer, . * Enrique Ramirez de Arellanon (1998) ''Operator theory for complex and hypercomplex analysis'',
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
(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, . * 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 ''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters pate ...
143:57–69. * 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. Functions and mappings Hypercomplex numbers Mathematical analysis