mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, hypercomplex analysis is the extension of

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

to the

hypercomplex number In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. ...

s. The first instance is functions of a quaternion variable, 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. The algebra of quater ...

(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 number In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...

s. In

mathematical physics Mathematical physics is 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 the de ...

, 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 with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...

s. 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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

of the

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

of hypercomplex numbers determines the function theory. Functions such as square root of a matrix,

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrix, 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 exp ...

, and logarithm of a matrix are basic examples of hypercomplex analysis. The function theory of diagonalizable matrices is particularly transparent since they have eigendecompositions.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 launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal complete ...

. Suppose

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

where the ''E''_''i'' are projections. Then for any

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

f

f(T) = \sum_^N  f(\lambda_i ) E_i.

The modern terminology for a "system of hypercomplex numbers" is an ''

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

over the real numbers'', and the algebras used in applications are often Banach algebras since

Cauchy sequence In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...

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 cal ...

s and series. 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 de ...

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. 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

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 or Florida State) is a Public university, public research university in Tallahassee, Florida, United States. It is a senior member of the State University System of Florida and a preeminent university in the s ...

* Sorin D. Gal (2004) ''Introduction to the Geometric Function theory of Hypercomplex variables'', Nova Science Publishers, . * * 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