Hypercomplex analysis

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, hypercomplex analysis is the extension of <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a> to the <a href="/facts/Hypercomplex_number/BwKwTBF2">hypercomplex numbers</a>. The first instance is functions of a <a href="/facts/Quaternion_variable/s6Gpj6ca">quaternion variable</a>, where the argument is a <a href="/facts/Quaternion/T3tnu7mX">quaternion</a> (in this case, the sub-field of hypercomplex analysis is called <a href="/facts/Quaternionic_analysis/s6Gpj6ca">quaternionic analysis</a>). A second instance involves functions of a <a href="/facts/Motor_variable/1LaC4wk2">motor variable</a> where arguments are <a href="/facts/Split-complex_number/DtavqChA">split-complex numbers</a>.
In <a href="/facts/Mathematical_physics/sLKg8T1G">mathematical physics</a>, there are hypercomplex systems called <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebras</a>. The study of functions with arguments from a Clifford algebra is called <a href="/facts/Clifford_analysis/L3ogVSyx">Clifford analysis</a>.
A <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> may be considered a hypercomplex number. For example, the study of functions of 2 × 2 <a href="/facts/Real_number/R02gw5Pb">real</a> matrices shows that the <a href="/facts/Topological_space/v2ut7CbK">topology</a> of the <a href="/facts/Space_(mathematics)/G6rt4bQ8">space</a> of hypercomplex numbers determines the function theory. Functions such as <a href="/facts/Square_root_of_a_matrix/uWGkQQqN">square root of a matrix</a>, <a href="/facts/Matrix_exponential/WID5cG2g">matrix exponential</a>, and <a href="/facts/Logarithm_of_a_matrix/lhmcCLJh">logarithm of a matrix</a> are basic examples of hypercomplex analysis. 
The function theory of <a href="/facts/Diagonalizable_matrices/y3MkyTCy">diagonalizable matrices</a> is particularly transparent since they have <a href="/facts/Eigendecomposition/a2nfF7hJ">eigendecompositions</a>. Suppose 
 
 
 
 
 T
 =
 
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 i
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 1
 
 
 N
 
 
 
 λ
 
 i
 
 
 
 E
 
 i
 
 
 
 
 
 {\displaystyle \textstyle T=\sum _{i=1}^{N}\lambda _{i}E_{i}}
 
 where the Ei are <a href="/facts/Projection_(linear_algebra)/sElXGkxD">projections</a>. Then for any <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
, 
 
 
 
 f
 (
 T
 )
 =
 
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 i
 =
 1
 
 
 N
 
 
 f
 (
 
 λ
 
 i
 
 
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 E
 
 i
 
 
 .
 
 
 {\displaystyle f(T)=\sum _{i=1}^{N}f(\lambda _{i})E_{i}.}
 
 
The modern terminology for a "system of hypercomplex numbers" is an <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> over the real numbers, and the algebras used in applications are often <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebras</a> since <a href="/facts/Cauchy_sequence/7Q5c8TuF">Cauchy sequences</a> can be taken to be <a href="/facts/Convergent_sequence/Ktge2RwL">convergent</a>. Then the function theory is enriched by <a href="/facts/Sequence/gV2S4uqp">sequences</a> and <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a>. In this context the extension of <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> of a <a href="/facts/Complex_number/2w7mqI7e">complex</a> variable is developed as the <a href="/facts/Holomorphic_functional_calculus/xtWI9f4V">holomorphic functional calculus</a>. Hypercomplex analysis on Banach algebras is called <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>.

Hypercomplex analysis open-in-new

Hypercomplex analysis