Operator calculus approach to solving analytic systems
Résumé
Solving analytic systems using inversion can be implemented in a variety of ways. One method is to use Lagrange inversion and variations. Here we present a different approach, based on dual vector fields. For a function analytic in a neighborhood of the origin in the complex plane, we associate a vector field and its dual, an operator version of Fourier transform. The construction extends naturally to functions of several variables. We illustrate with various examples and present an efficient algorithm readily implemented as a symbolic procedure in Maple while suitable as well for numerical computations using languages such as $C$ or Java.