Monogenic Function

Monogenic Function

 

a function of a complex variable that is defined on a given set and that has a derivative at some point on the set (monogenic function at a point) or at every point on the set (monogenic function on a set). Whenever the given set is a domain, the concept of a monogenic function on a set coincides with the concept of an analytic function.

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The fundamental references for special monogenic function are [36, 37].
The basic set [([P.sub.k]).sub.k [greater than or equal to] 0] will be effective for H(r) in [bar.H](R) if the basic series of each special monogenic function in B(r) converges to f normally in [bar.B](R).
An important subclass of the Clifford holomorphic functions called special monogenic functions is considered, for which a Cannon theorem on the effectiveness in closed and open ball [21,23] was established.
It is well known that [([P.sub.n](x)).sub.n [greater than or equal to] 0] is an Appell sequence with respect to [partial derivative]/[partial derivative][x.sub.0] or (1/2)D (which represent the same operator for monogenic functions): (1/2)[bar.D][P.sub.n](x) = n[P.sub.n-1](x) and [bar.D] = [[summation].sup.m.sub.i=0][[bar.e].sub.i]([partial derivative]/[partial derivative][x.sub.i]) in [R.sup.m+1] (see [38-40]).
Also, the class of special monogenic functions in an open ball B(r) is written as H(r) and [bar.H](r) denotes the class of special monogenic functions in closed ball [bar.B](r).
The [A.sub.m]-module F of Section 6 will be taken as the class [bar.H](R) of special monogenic functions in the closed ball [bar.B](R), with the norm [sigma] defined by
Let us mention that holomorphic functions of several complex variables are a special case of Hermitian monogenic functions. In the recent papers [1, 2, 3, 4, 10, 11], the Hermitean Clifford analysis setting was further developed by following a circulant (2 x 2) matrix framework.
To the best of our knowledge, a Carleman-type theorem has never proved neither for Cauchy-Fueter regular functions of a quaternionic variable nor for monogenic functions with values in a Clifford algebra.
The most studied notion of (hyper)holomorphy in this setting leads to the so-called monogenic functions, see [1].
Subharmonicity of powers of octonion-valued monogenic functions and some applications.