A website about algebraic functions and iterated exponential and polynomial systems
$$ \newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} \newcommand{\wvalx}{\underbrace{z^{\lambda_4}(c_4+w_5)}_{w_4}} \newcommand{wvalxx}{\underbrace{z^{\lambda_3}(c_3+\wvalx)}_{w_3}} \newcommand{wvalxxx}{\underbrace{z^{\lambda_2}\{c_2+\wvalxx\}}_{w_2}} \newcommand{wvalxxxx}{z^{\lambda_1}\big(c_1+\wvalxxx\big)} $$An algebraic function $w(z)$ is given implicitly by the expression $$ \begin{equation} f(z,w)=a_0(z)+a_1(z)w+a_2(z)w^2+\cdots+a_n(z)w^n=0 \label{eqn001} \end{equation} $$ with $z$ and $w$ complex variables and the coefficients, $a_i(z)$, polynomials in $z$ with rational coefficients; and iterated exponential and polynomial systems. Readers are advised to read the indicated background sections in order to better understand the content of each section.
The software used in this web site is Mathematica.
Algebraic functions:
Readers interested in learning more about algebraic functions may wish to look for my upcoming book Algebraic Functions, A Computational Introduction using Mathematica covering the information about algebraic functions in this web site in more detail as well as other subjects such as genus and analytic continuation of the Beta, Zeta, and Gamma functions and will include 13 freely-available Mathematica notebooks to assist with the computational discovery. The book is a 500-page, full-color, spiral-bound, two-volume set. It is currently in review and should be available by September, 2026.
- Section 0: Preliminaries
- Section 1: Introduction
- Section 2: An Improved Plotting Method
- Section 3: Applying Laurent's Theorem to Algebraic Functions
- Section 4: Applying the Residue Theorem to Algebraic Functions
- Section 5: Mathematica Code
- Section 6: Puiseux Series (background)
- Section 7: Puiseux Series (examples)
- Section 8: Designing doPuiseux
- Section 9: Finite power series (polynomials)
- Section 10: Region of Convergence of annular power expansions of Algebraic functions
- Section 11: Riemann Surfaces
- Section 12: Evaluating the Indeterminant Form
- Section 13: Analyzing the Annular Laurent Integrals
- Section 14: Analyzing the Annular Laurent Puiseux Series
Puiseux expansions around singular points and their radii of convergence:
- Section 15: Determing radii of convergence of Puiseux power expansions around singular points of algebraic functions
- Section 16: Radius of convergence part II
- Section 17: Radius of convergence Part III: Some extreme examples
Iterated exponential functions:
- Section A: Introduction to fixed points of iterated exponentials
- Section B: Computing the branching parameters of iterated exponentials
