On complex singularity analysis for some linear partial differential equations in C

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in C outside some singular set Θ. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea we have initiated in a previous work [20]. The solutions Y are shown to develop singularities along Θ with estimates of exponential type depending on the growth’s rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.