Cauchy Integral Decomposition of Multi-Vector Valued Functions on ...

118 R. Abreu Blaya, J. Bory Reyes, R. Delanghe and F. Sommen CMFT
Example. An important example is the case <strong>of</strong> the unit sphere S m−1 in R m .
Let us describe ∂ ‖x as an operator acting from the left. Using polar coordinates
x = rξ with r = |x| and ξ ∈ S m−1 , we may write ∂ x as (see [5])
(
∂ x = ξ ∂ r + Γ )
ξ
r
where
Γ ξ = x ∧ ∂ x .
As at ξ ∈ S m−1 , n(ξ) = ξ, we thus have in terms <strong>of</strong> polar coordinates that in
Σ ɛ = S m−1 ×] − ɛ, ɛ[, ɛ sufficiently small,
∂ ‖x = ξΓ ξ
r ,
its restriction ∂ ω to S m−1 being given by
∂ ω = ξΓ ξ .
In the case m = 2, i.e. the case <strong>of</strong> the unit circle S 1 in the plane, straightforward
computations then lead to
where
is the unit tangent vector at ξ ∈ S 1 .
∂ ‖x = T (ξ)∂ θ
,
r
∂ ω = T (ξ)∂ θ ,
( π
) ( π
)
T (ξ) = e 1 cos
2 + θ + e 2 sin
2 + θ
2.3. Some elements <strong>of</strong> Clifford analysis. Let G be an open subset <strong>of</strong> R m
and let f : G → R 0,m be a C 1 -function. Then f is said to be left (resp. right)
monogenic in G if ∂ x f = 0 in G (resp. f∂ x = 0 in G). If f is both left and right
monogenic in G, i.e. ∂ x f = f∂ x = 0 in G, then f is called two-sided monogenic
in G. Monogenic functions in G belong to C ∞ (G); even more, they are real
analytic in G (see e.g. [5]). The space <strong>of</strong> left monogenic functions and <strong>of</strong> twosided
monogenic functions in G is denoted, respectively, by M(G) and M(G).
An important example <strong>of</strong> a two-sided monogenic function in G = R m \ {0} is
given by the fundamental solution E <strong>of</strong> ∂ x , namely
E(x) = 1 x
A m |x| . m
Here A m stands for the surface area <strong>of</strong> the unit sphere S m−1 in R m .
Notice that if F k is an R (k)
0,m-valued C 1 -function in G (F k is also called k-multivector
valued, 0 ≤ k ≤ m), then in G
∂ x F k = 0 ⇐⇒ F k ∂ x = 0.