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Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483801.png" /> of the hypercomplex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483802.png" /> (cf. [[Hypercomplex number|Hypercomplex number]]) over the field of real numbers, i.e. a function on a finite-dimensional associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483803.png" />. In a restricted sense, a hypercomplex function is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483804.png" /> with values in the same algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483805.png" />, i.e. the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483806.png" /> may be represented as
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483807.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h0483809.png" />, is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838010.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838012.png" />, is a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838013.png" /> real functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838014.png" /> real variables. The theory of hypercomplex functions has been most thoroughly studied for quaternion algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838015.png" /> (cf. [[Quaternion|Quaternion]]).
+
Functions  $  w ( z) $
 +
of the hypercomplex variable  $  z $(
 +
cf. [[Hypercomplex number|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|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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838017.png" />-analytic hypercomplex functions, [[#References|[1]]]. A hypercomplex function is said to be right regular at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838018.png" /> if the differential equation (Fueter's condition)
+
In modern studies principal stress is laid on regular hypercomplex functions analytic according to Fueter, or $  F $-
 +
analytic hypercomplex functions, [[#References|[1]]]. A hypercomplex function is said to be right regular at a point $  z _ {0} $
 +
if the differential equation (Fueter's condition)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838019.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 0 } ^ { {n }  - 1 } w  ^ {(} k) e _ {k}  = 0,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838020.png" /></td> </tr></table>
+
$$
 +
w  ^ {(} k)  = \
 +
\sum _ {h = 0 } ^ { {n }  - 1 }
 +
 
 +
\frac{\partial  u _ {h} }{\partial  x _ {k} }
 +
  e _ {h}  $$
  
is the partial derivative of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838021.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838022.png" />, is satisfied at that point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838023.png" />; all derivatives are assumed to be continuous. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838024.png" /> is said to be a left-regular hypercomplex function if
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838025.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 0 } ^ { {n }  - 1 } e _ {k} w  ^ {(} k)  = 0.
 +
$$
  
In the case of a non-commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838026.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838029.png" /> (for left-regular hypercomplex functions — of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838030.png" />); hence a specific integral theorem is obtained.
+
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 [[#References|[2]]] at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838031.png" /> for the case of a commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838032.png" /> is a hypercomplex function for which the differential at that point may be written as
+
A hypercomplex function analytic according to Scheffers [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838033.png" /></td> </tr></table>
+
$$
 +
dw  = \phi ( z)  dz,
 +
$$
  
where the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838034.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838035.png" />. In the case of a commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838036.png" /> this condition is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838037.png" />, and the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838038.png" /> is independent of the path. Hypercomplex functions that are analytic according to Scheffers are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838039.png" />-regular if and only if
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838040.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 0 } ^ { {n }  - 1 } e _ {k}  ^ {2}  = 0.
 +
$$
  
A hypercomplex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838041.png" /> is said to be analytic according to Hausdorff [[#References|[3]]] at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838042.png" /> if its differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838043.png" /> is a linear function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838044.png" />, i.e. if
+
A hypercomplex function $  w ( z) $
 +
is said to be analytic according to Hausdorff [[#References|[3]]] at a point $  z _ {0} $
 +
if its differential $  dw $
 +
is a linear function of $  dz $,  
 +
i.e. if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838045.png" /></td> </tr></table>
+
$$
 +
dw  = \
 +
\sum _ {i, k = 0 } ^ { {n }  - 1 } \phi _ {ik} e _ {i}  dz  e _ {k} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838046.png" /> are real functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838047.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048380/h04838048.png" /> Hausdorff's and Scheffers' definitions are equivalent.
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Fueter,  "Ueber die Funktionentheorie in einer hyperkomplexen Algebra"  ''Elemente der Math.'' , '''3''' :  5  (1948)  pp. 89–94</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Scheffers,  "Verallgemeinerung der Grundlagen der gewöhnlich complexen Funktionen"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Mat.-Phys. Kl.'' , '''45'''  (1893)  pp. 828–848</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Hausdorff,  "Zur Theorie der Systeme complexer Zahlen"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Mat.-Phys. Kl.'' , '''52'''  (1900)  pp. 43–61</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.Kh. Kristalinskii,  "Pseudoregular quaternion functions"  ''Uchen. Zap. Smolensk. Ped. Inst.'' , '''14'''  (1965)  pp. 91–95  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Fueter,  "Ueber die Funktionentheorie in einer hyperkomplexen Algebra"  ''Elemente der Math.'' , '''3''' :  5  (1948)  pp. 89–94</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Scheffers,  "Verallgemeinerung der Grundlagen der gewöhnlich complexen Funktionen"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Mat.-Phys. Kl.'' , '''45'''  (1893)  pp. 828–848</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Hausdorff,  "Zur Theorie der Systeme complexer Zahlen"  ''Ber. Verh. Sächs. Akad. Wiss. Leipzig Mat.-Phys. Kl.'' , '''52'''  (1900)  pp. 43–61</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.Kh. Kristalinskii,  "Pseudoregular quaternion functions"  ''Uchen. Zap. Smolensk. Ped. Inst.'' , '''14'''  (1965)  pp. 91–95  (In Russian)</TD></TR></table>

Latest revision as of 22:11, 5 June 2020


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)
How to Cite This Entry:
Hypercomplex functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypercomplex_functions&oldid=13128
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article