A website about algebraic functions and iterated exponential and polynomial systems
$$ \newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} \newcommand{\wvalx}{\underbrace{z^{\lambda_4}(c_4+w_5)}_{w_4}} \newcommand{wvalxx}{\underbrace{z^{\lambda_3}(c_3+\wvalx)}_{w_3}} \newcommand{wvalxxx}{\underbrace{z^{\lambda_2}\{c_2+\wvalxx\}}_{w_2}} \newcommand{wvalxxxx}{z^{\lambda_1}\big(c_1+\wvalxxx\big)} $$Obtaining the free PDF book Algebraic Functions: A Computational Introduction Using Mathematica from GitHub
This book provides a detailed introduction to algebraic functions and includes a full set of companion Mathematica notebooks. To access the materials, follow these steps:
- Go to github.com and search for the algFunctionBook repository. This will bring up the GitHub repository containing the book and display the README file.
- Read the README file carefully. To view the book PDF directly, click the PDF folder and then select algebraicFunctionBook.pdf. GitHub will display a reduced‑resolution preview without clickable links.
- To obtain the full-resolution PDF, click the small download icon (the “arrow into a tray”) in the upper right of the preview. Firefox is the recommended viewer because it supports reliable back‑navigation from hyperlinks using Alt + Left Arrow, but any PDF viewer with good navigation will work.
- Readers who want the complete computational experience can follow the instructions in the README file and download notebookLoader.nb from the notebook folder. Open it in Mathematica 13 or later and run the loader. It will automatically download the chapter notebooks and the PDF into the directory structure described in README.
Algebraic functions:
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.
- 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
