We show that solutions of analytic elliptic partial differential
equations of the form
formula here
in a simply connected domain Ω ⊂ ℝn
can be extended holomorphically into
[Nscr ] (Ω) ⊂ [Copf ]n, the so-called
kernel of the harmonicity hull of Ω. This extends results of
Avanissian, Aronszajn etc., on polyharmonic
functions and also results of Vekua, Khavinson and Shapiro in
ℝ2. We also find the domain of influence
for solutions of a certain subclass of these operators, in terms of
their Cauchy data on analytic
hypersurfaces in [Copf ]n (a complex Huygens'
principle). As an application, we investigate reflection properties
of these solutions and, in particular, solutions of the Helmholtz
equation in ℝ3.