Hypercomplex functions

Functions $ w ( z) $ of the hypercomplex variable $ z $( cf. Hypercomplex number) over the field of real numbers, i.e. a function on a finite-dimensional associative algebra $ \mathfrak A $. In a restricted sense, a hypercomplex function is a function $ w ( z) $ with values in the same algebra $ \mathfrak A $, i.e. the function $ w( z) $ may be represented as

$$ w ( z) = \ \sum _ {k = 0 } ^ { {n } - 1 } e _ {k} u _ {k} , $$

where $ e _ {k} $, $ k = 0 \dots n - 1 $, is a basis of $ \mathfrak A $, while $ u _ {k} = u _ {k} ( x _ {0} \dots x _ {n-} 1 ) $, $ k = 0 \dots n - 1 $, is a system of $ n $ real functions in $ n $ real variables. The theory of hypercomplex functions has been most thoroughly studied for quaternion algebras $ \mathfrak A $( cf. Quaternion).

Analytic (regular) hypercomplex functions are generalizations in different directions of analytic functions of one complex variable. The concepts of an analytic hypercomplex function differ, on account of the fact that the definitions of analyticity in arbitrary algebras need not be equivalent.

In modern studies principal stress is laid on regular hypercomplex functions analytic according to Fueter, or $ F $- analytic hypercomplex functions, [1]. A hypercomplex function is said to be right regular at a point $ z _ {0} $ if the differential equation (Fueter's condition)

$$ \sum _ {k = 0 } ^ { {n } - 1 } w ^ {(} k) e _ {k} = 0, $$

where

$$ w ^ {(} k) = \ \sum _ {h = 0 } ^ { {n } - 1 } \frac{\partial u _ {h} }{\partial x _ {k} } e _ {h} $$

is the partial derivative of the function $ w $ with respect to $ x _ {k} $, is satisfied at that point $ z _ {0} $; all derivatives are assumed to be continuous. A function $ w( z) $ is said to be a left-regular hypercomplex function if

$$ \sum _ {k = 0 } ^ { {n } - 1 } e _ {k} w ^ {(} k) = 0. $$

In the case of a non-commutative algebra $ \mathfrak A $ these concepts are not equivalent. The sum and the difference of right-regular hypercomplex functions are right regular, but this is not true for their product or quotient. Powers of the variable $ z $ are not right-regular. There exist Taylor and Laurent series for specially constructed analogues of powers. Fueter's condition is equivalent to the vanishing of the differential of the hypercomplex differential form $ \omega = w dz $, $ \delta \omega = 0 $( for left-regular hypercomplex functions — of the form $ \omega = dz w $); hence a specific integral theorem is obtained.

A hypercomplex function analytic according to Scheffers [2] at a point $ z _ {0} $ for the case of a commutative algebra $ \mathfrak A $ is a hypercomplex function for which the differential at that point may be written as

$$ dw = \phi ( z) dz, $$

where the derivative $ \phi ( z) = dw/dz $ is independent of $ dz $. In the case of a commutative algebra $ \mathfrak A $ this condition is equivalent to $ d \omega = 0 $, and the integral $ \int w dz $ is independent of the path. Hypercomplex functions that are analytic according to Scheffers are $ F $- regular if and only if

$$ \sum _ {k = 0 } ^ { {n } - 1 } e _ {k} ^ {2} = 0. $$

A hypercomplex function $ w ( z) $ is said to be analytic according to Hausdorff [3] at a point $ z _ {0} $ if its differential $ dw $ is a linear function of $ dz $, i.e. if

$$ dw = \ \sum _ {i, k = 0 } ^ { {n } - 1 } \phi _ {ik} e _ {i} dz e _ {k} , $$

where $ \phi _ {ik} $ are real functions in $ x _ {0} \dots x _ {n-} 1 $. In this case the analogues of power series are easier to construct, but the value of the integral depends on the path. For a commutative algebra $ \mathfrak A $ Hausdorff's and Scheffers' definitions are equivalent.

References

[1]	R. Fueter, "Ueber die Funktionentheorie in einer hyperkomplexen Algebra" Elemente der Math. , 3 : 5 (1948) pp. 89–94
[2]	G. Scheffers, "Verallgemeinerung der Grundlagen der gewöhnlich complexen Funktionen" Ber. Verh. Sächs. Akad. Wiss. Leipzig Mat.-Phys. Kl. , 45 (1893) pp. 828–848
[3]	F. Hausdorff, "Zur Theorie der Systeme complexer Zahlen" Ber. Verh. Sächs. Akad. Wiss. Leipzig Mat.-Phys. Kl. , 52 (1900) pp. 43–61
[4]	R.Kh. Kristalinskii, "Pseudoregular quaternion functions" Uchen. Zap. Smolensk. Ped. Inst. , 14 (1965) pp. 91–95 (In Russian)