Hypercomplex analysis

In mathematics, hypercomplex analysis is the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number. The first instance is functions of a quaternion variable, where the argument is a quaternion. 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, 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. The function theory of diagonalizable matrices is particularly transparent since they have eigendecompositions. Suppose ${\displaystyle \textstyle T=\sum _{i=1}^{N}\lambda _{i}E_{i}}$ where the E_i are projections. Then for any polynomial ${\displaystyle \textstyle f,\quad f(T)=\sum _{i=1}^{N}f(\lambda _{i})E_{i}.}$

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