If $|a_{mn}x_0^my_0^n| \leq M$ then a double power series $f(x,y) = \sum a_{mn} x^m y^n$ can be 'bounded' by a dominant function of the form $\phi(x,y) = \tfrac{M}{(1-\tfrac{x}{x_0})(1-\tfrac{y}{y_0})}$, obviously derived from a geometric series argument. This is useful when proving that analytic solutions exist to $y' = f(x,y)$ in the case that $f$ is analytic.
A more useful dominant function when proving existence for an integrable pfaffian of the form $dz = f_1dx_1 + f_2dx_2$ is given by $\psi(x,y) = \tfrac{M}{1-(\tfrac{x}{x_0}+\tfrac{y}{y_0})}$. The coefficients of $x^my^n$ in the taylor expansion of $\psi$ is equal to the coefficients of $x^my^n$ in the taylor expansion of $M(\tfrac{x}{x_0}+\tfrac{y}{y_0})^{m+n}$, and are at least equal to the coefficients of the taylor expansion of $\phi$.
My question is, how in the world does one gain any intuition for all of this? I can understand, derive & use $\phi(x,y) = \tfrac{M}{(1-\tfrac{x}{x_0})(1-\tfrac{y}{y_0})}$ perfectly, however motivating, deriving & using the alternative function $\psi(x,y) = \tfrac{M}{1-(\tfrac{x}{x_0}+\tfrac{y}{y_0})}$ is too arbitrary for me, is there a nice way to see the use of this second dominant function as obvious? Page 397 if needed
