On Regular Functions of a Nonalternative Hypercomplex Variable

概要

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For the algebra of hypercomplex numbers in our theory, neither the associative law nor the alternative law is assumed. We have assumed the power associative law for the algebra and the anticommutation relations for the units i_κ (k=1,…, 2s-1) : i_1i_κ+i_κi_i=-2δ_<κ1>. There are several subvariables which do not contain certain coordinate variables x_κ. Accordingly, there are several regularity conditions each of which is applicable to functions of a specific variable. It is shown that a function G(X)≡□^<s-1>F(X) is a regular function of a hypercomplex variable X=x_0+i_1x_1+…+i_<2s-1>x_<2s-1>, [numerical formula] and assuming that F(X) is a regular function of a complex variable Χ=x_0+ix. We have derived the integral theorems, regular polynomial functions, exponential functions and Fourier integral theorems. The results may be of use even to those functions of a Clifford variable and to those of an octonion variable which are alternative or even associative.
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