An interesting case of variables separation (Q1585522)

There is an application of holomorphic expansions \[ W= \sum^\infty_{|n|= 0}\overline z^n W_n(z)\tag{HE} \] and connected results to construct explicit solution for linear partial differential equation (PDE) with constant coefficients, as for instance heat, wave or Laplace equations. The method consist of writing the complex analog of PDE and search for a solution in form of an (HE) with unknown coefficients \(W_n(z)\). The theoretical justification for this technique is done. As an example, a particular solution for a nonhomogeneous wave equation is computed.