  The Weinstein equation with complex coefficients is the equation governing
generalized axisymmetric potentials (GASP) which can be written as
$L_m[u]=\Delta u+\left(m/x\right)\partial_x u =0$, where $m\in\mathbb{C}$. We
generalize results known for $m\in\mathbb{R}$ to $m\in\mathbb{C}$. We give
explicit expressions of fundamental solutions for Weinstein operators and their
estimates near singularities, then we prove a Green's formula for GASP in the
right half-plane $\mathbb{H}^+$ for Re $m<1$. We establish a new decomposition
theorem for the GASP in any annular domains for $m\in\mathbb{C}$, which is in
fact a generalization of the B\^ocher's decomposition theorem. In particular,
using bipolar coordinates, we prove for annuli that a family of solutions for
GASP equation in terms of associated Legendre functions of first and second
kind is complete. For $m\in\mathbb{C}$, we show that this family is even a
Riesz basis in some non-concentric circular annulus.
