mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 converg ...

and

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

to the study of functions where the

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

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 represent ...

. The first instance is functions of a

quaternion variable In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. ...

, where the argument is a

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

(in this case, the sub-field of hypercomplex analysis is called

quaternionic analysis In mathematics, quaternionic analysis is the study of function (mathematics), functions with quaternions as the domain of a function, domain and/or range. Such functions can be called functions of a quaternion variable just as Function of a real va ...

). A second instance involves functions of a

motor variable In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor fo ...

where arguments are

split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...

s. In

mathematical physics Mathematical physics refers to the development of mathematics, 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 t ...

, there are hypercomplex systems called

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...

s. 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 a ...

. A

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

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 language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

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 consider ...

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 gives ...

, 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 In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...

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 matri ...

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 reference ...

. Suppose

\textstyle T = \sum _^N \lambda_i E_i

where the ''E''_''i'' are

projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...

s. 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 exa ...

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 a ...

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 spa ...

s since

Cauchy sequence In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

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 i ...

. In this context the extension of

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

s 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''(''T ...

. 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 space#Definition, inner product, Norm (mathematics)#Defini ...

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 Florida State University (FSU) is a public research university in Tallahassee, Florida. It is a senior member of the State University System of Florida. Founded in 1851, it is located on the oldest continuous site of higher education in the st ...

* 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 for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure a ...

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

See also

References

Sources