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Octonion Functions:

Some early attempts to a straightforward extension of the theory of functions of a complex variable or of the theory of functions of several variables, even to a 4-dimensional case of quaternions, did not yield interesting functions.  Over the years many researchers developed extensions of the theory of regular (monogenic) functions, that was constructed by Fueter for quaternions, which shown to have some applications.  Here is a simple description of one such construction for an octonion variable.

Define an octonion differential operator, D, by

D :=  m=0S7
 d       em  . dxm
Since the octonion multiplication is non-commutative it is necessary to distinguish between the left and the right multiplication of a function by the differential operator, D.

Definition :
Let F(X) be a function of an octonion variable, X, such that F(X) can be expressed as
F(X) = m=0S7  fm  em,
where each of fm=  fm( x0 , x1 ,..., x7 ) ,    m  = 0, 1, ...,7,
is a continuous and differentiable function (in the usual sense of the first year calculus) of x0 , x1 ,..., x7 , in a certain domain, U. Then F(X) is left regular at X in U if
D F(X)  =  0.
Similarly, F(X) is right regular at X in U if
F(X) D  =  0.

Example:
Let g(z) = u ( x0, x ) +  i v ( x0, x )  be an analytic function of a complex variable,  z = x0 + i x , where i2 =  - 1.
Replace

i   ----»
 [  j=1S7  xj ej   ] ___________________    , [  ( j=1S7  xj2 ) 1/2  ]
and x  ----» ( j=1S7  xj2 ) 1/2 .  Then the function assumes the form

G(X)  =  u ( x0, x )  +
 [ j=1S7  v ( x0, x )  xj ej ] _______________________  . [  ( j=1S7  xj2 ) 1/2  ]
Now, define an octonion function
F(X) :=  (DD*) G(X) ,   where D* is a conjugate of D, and

DD* =     m=0S7
 d   2 dxm2
Then F(X) is both left and right regular function.

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