On a generalized taylor theorem: A rational proof of the validity of the homotopy analysis method

Abstract

A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can he applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.