The Weinstein equation with complex coefficients is the equation governing axisymmetric potentials (PSA) which can be written as $L_m[u]=\Delta u+\left(m/x\right)\d_x u =0$, where $m\in\C$. This equation is used in particular for modeling the plasma shape in a Tokamak (toroidal chamber with axial magnetic field) for $m = -1$, or it is, when $m=1$, the well-known linearized Ernst equation (which is used to give explicit solutions of the Einstein equations). Here, we generalize results known for $m\in\R$ to $m\in\C$. We give explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities, then we prove a Green's formula for PSA in the right half-plane $\H^+$ for Re $m<1$. We establish a new decomposition theorem for the PSA in any annular domains for $m\in\C$. In particular, using bipolar coordinates, we prove for annuli (always for $m\in\C$) that a family of solutions for PSA equation in terms of associated Legendre functions of first and second kind is complete (the method rests on quasi-separability of variables and some Fourier analysis). For $m\in\R$, we show that this family is even a Riesz basis in some non-concentric circular annulus. In the second part, basing on a method due to A. S. Fokas, we give, in explicit integral form, formulas for PSA in a circular domain of the right-half plane $\H^+$ when $m$ is an integer. These representations are obtained by solving a Riemann-Hilbert problem on the complex plane or on a Riemann surface with two sheets according to the parity of $m$. These formulas involve in an explicit form the Dirichlet and the Neumann data of the PSA in question. In the last part, we study a class of functions which includes the PSA, namely the pseudo-holomorphic functions, i.e. solutions of the complex equation $\overline{\partial} w=\alpha \overline{w}$, with $\alpha\in L^r$, $2\leq r<\infty$. We extend the Bers similarity principle (decomposition of pseudo-holomorphic functions in the form $e^s \,F$ under some regularity assumptions with holomorphic $F$) and a converse of this principle to the critical regularity case $r=2$. Using the connection between pseudo-holomorphic functions and solutions to the conjugate Beltrami equations, we deduce well-posedness of Dirichlet problem in smooth domains with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in the Sobolev space $W^{1,2}$.
