Lately I care a lot about the singularities of the solutions of PDEs. As a simple example I looked at the equation $$ u_{zz}(z,s) + u_{ss}(z,s) = f(z,s). $$ With $f$ analytic in $\mathbb{C}^2$. (It is also of exponential type, but I am not sure that this is really relevant) Forgive me for the non-standard variables, but for some reasons I ended up using these.
I don't really know much about analytic PDEs, but as far as I understand the idea of boundary values in this setting is replaced by the singularities of the solution. So the above equation is actually well-defined in this setting. Am I correct?
Since it is a linear wave equation (one would say elliptic but in $\mathbb{C}$ this does not really make a lot of sense) I expect that it is easily solved. Is there a well known general solution?
Playing with the Hadamard product I found out (hopefully correctly) that $u$ is singular when $s=\pm i z$. I hope I can retrieve this (and more) information from the general solution.