Functional Power Series

TL;DR

This paper introduces a versatile functional power series expansion method for analytic functions, providing a new approach for approximations and derivatives, including composite and inverse-composite functions.

Contribution

It presents a novel functional series expansion applicable to any analytic function, along with new formulas for derivatives of composite functions and inverse-composite functions.

Findings

Effective for approximating functions with few terms

Provides new expressions for composite function derivatives

Handles inverse-composite functions efficiently

Abstract

This work introduces a new functional series for expanding an analytic function in terms of an arbitrary analytic function. It is generally applicable and straightforward to use. It is also suitable for approximating the behavior of a function with a few terms. A new expression is presented for the composite function's n'th derivative. The inverse-composite method is handled in this work also.