On the algebraic fundamentals of singular perturbation theory (Q354533)

The aim of this paper is to provide a generalization of the Poincaré expansion theorem to the Cauchy problem for singularly perturbed systems of the form \[ \epsilon\frac{dy}{dx}=f(x,y),\qquad y(x_0,\epsilon)=y_0 \] in the complex plane. The holomorphic regularization method is applied to construct first integrals of the singularly perturbed systems under consideration. The existence of a unique solution \(y(x,\epsilon)\), pseudoholomorphic in the global sense, is proven.