Abstract
In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function; in other words, given y=f(x), we provide the power series that defines its inverse x=hf(y). We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss–Catalan numbers.
Highlights
In this paper, we have taken as a basis the previous works [1,2], in which the inverse of a polynomial function is constructed, with the aim of generalizing the methods developed there to any analytic function
In the two sections, we introduce some applications for solving nonlinear equations in an analytic way, and to generalize the Catalan and Fuss–Catalan numbers
Due to the Inverse Function Theorem, given an analytic real function, f ( x ), and a point, x0, with f 0 ( x0 ) 6= 0, there is a neighborhood of x0, Vx0, in such a manner that the inverse function of f ( x ) is well defined in f (Vx0 )
Summary
We have taken as a basis the previous works [1,2], in which the inverse of a polynomial function is constructed, with the aim of generalizing the methods developed there to any analytic function. In the two sections, we introduce some applications for solving nonlinear equations in an analytic way, and to generalize the Catalan and Fuss–Catalan numbers. Catalan numbers appeared for the first time in the book Quick Methods for Accurate Values of Circle Segments, by Ming Antu (1692–1763), a Chinese mathematician. In this book, he provides some trigonometric equalities and power series, in which Catalan numbers are involved. Throughout this paper, all the necessary computational tasks have been performed with the program Wolfram Mathematica 11.2.0.0
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
