Some Power Series Expansions for Monogenic Functions in

266 D. Peña Peña and F. Sommen CMFT
We must remark that despite the fact that Clifford analysis generalizes the most
important features of classical complex analysis, the monogenic functions do not
enjoy all the properties of the holomorphic functions of one complex variable. For
instance, the product of two monogenic functions is not necessarily monogenic
because of the non-commutativity of the Clifford algebras. It is then natural to
look for techniques to construct monogenic functions.
There are a lot of techniques available to generate monogenic functions (a list of
those techniques can be found e.g. in [4]). This paper is related with two of these
techniques: the so-called axial monogenic functions and the Cauchy-Kowalewski
extension problem.
Let P k (x) be a homogeneous left monogenic polynomial of degree k in R m . The
axial monogenic functions (see [6, 12]) are monogenic functions of the form
(A(x 0 , r) + ω B(x 0 , r)) P k (x), r = |x|, rω = x,
A and B being scalar functions satisfying the Vekua-type system
∂ x0 A − ∂ r B = 2k + m − 1 B
r
∂ x0 B + ∂ r A = 0.
For more references on axial monogenic functions and special monogenic functions,
we refer to [8, 14, 15] and for the general theory of Vekua systems, we refer
to [1, 17].
The Cauchy-Kowalewski extension problem consists in finding a monogenic extension
g ∗ of a real-analytic function g defined on a given subset of R m+1 (see
e.g. [2, 3, 5, 6, 7, 11, 13, 16]).
If the given subset is an analytic m-surface in R m+1 , then in [13] Sommen proved
the existence and uniqueness of such extension. In that paper he improved the
classical Cauchy-Kowalewski extension formula obtained in [2] for functions defined
on the plane {(x 0 , x) ∈ R m+1 : x 0 = 0}. The classical Cauchy-Kowalewski
extension formula is given by the following series
∞∑
(1) g ∗ (−x 0 ) k
(x 0 , x) =
∂ k
k!
xg(x).
k=0
The Cauchy-Kowalewski extension problem has not yet been solved for surfaces
with higher codimension, except in the flat case (see [5, 6]). In this paper we
present possible methods related to solving this problem for special subsets of
R m+1 such as: spheres, cylinders and planes. Our methods also provide special
solutions of the above mentioned Vekua systems.
2. Preliminaries
Let m ∈ N and let {e 1 , . . . , e m } be an orthonormal basis of R m , then a basis
for the Clifford algebra R 0,m or its complexification C m = R 0,m ⊗ C is given by