Some Power Series Expansions for Monogenic Functions in

268 D. Peña Peña and F. Sommen CMFT
while the generalized Cauchy-Riemann operator D x = ∂ x0 +∂ x splits the Laplace
operator in R m+1 ∆ = ∂ 2 x 0
+ ∆ x = (∂ x0 + ∂ x )(∂ x0 − ∂ x ).
A continuously differentiable C m -valued function g defined in some open set of
R m (R m+1 respectively) satisfying the equation ∂ x g = 0 (D x g = 0 respectively)
is called a left monogenic function. In what follows we will say “f is monogenic”
rather than “f is left monogenic”.
Passing to spherical coordinates x = rω, r = |x| and ω ∈ S m−1 , S m−1 being the
unit sphere in R m , we have
(
∂ x = ω ∂ r + 1 )
r Γ x ,
where
Γ x = −x ∧ ∂ x = − ∑ j