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 4dimensional 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
Since the octonion multiplication is noncommutative 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=0}S^{7}
f_{m} e_{m,}
where each of f_{m}=
f_{m}( x_{0}
, x_{1} ,..., x_{7}
) , m = 0, 1, ...,7,
is a continuous and differentiable function (in the usual
sense of the first year calculus) of x_{0}
, x_{1} ,..., x_{7}
, 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 ( x_{0}, x
) + i v ( x_{0},
x ) be an analytic function of a complex variable, z = x_{0
}+
i x , where i^{2} =  1.
Replace
i » 
[ _{j=1}S^{7}
x_{j} e_{j}
] 
___________________ , 
[ ( _{j=1}S^{7}
x_{j}^{2 }) ^{1/2}
] 

and x » ( _{j=1}S^{7}
x_{j}^{2 }) ^{1/2}
. Then the function assumes the form
G(X) = u ( x_{0},
x ) + 
[ _{j=1}S^{7}
v ( x_{0}, x ) x_{j}
e_{j} ] 
_______________________ . 
[ ( _{j=1}S^{7}
x_{j}^{2 }) ^{1/2}
] 

Now, define an octonion function
F(X) :=
(DD*)^{3 }G(X)
, where D* is a conjugate of D, and
Then F(X) is both left and right regular function. 