A website about algebraic functions and iterated exponential and polynomial systems
$$ \newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} \newcommand{\wvalx}{\underbrace{z^{\lambda_4}(c_4+w_5)}_{w_4}} \newcommand{wvalxx}{\underbrace{z^{\lambda_3}(c_3+\wvalx)}_{w_3}} \newcommand{wvalxxx}{\underbrace{z^{\lambda_2}\{c_2+\wvalxx\}}_{w_2}} \newcommand{wvalxxxx}{z^{\lambda_1}\big(c_1+\wvalxxx\big)} $$This web site is about algebraic functions $w(z)$ 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:
- 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: Radius of Convergence of Algebraic Power Series
- 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
An algebraic function is a function that satisfies were and is a polynomial. with integer coefficients. Functions that can be constructed using only a finite number of elementary operations, as well as inverse algebraic functions of functions capable of being constructed in this way, are examples. There are many different types of algebraic functions: linear, quadratic, cubic, polynomial, rational, and radical equations. To learn algebraic functions fast your basics should be clear.
ReplyDeleteNoted. Thanks. I restricted the definition to the type of functions covered and given by (1) in the home page.
DeleteDear dominic,
ReplyDeleteI think in section 6 (Puiseux Series) in formula (1)
the first equation should be \mu + ord(a_n) = n\lambda and not \mu + ord(a_i) = n\lambda
Best wishes
Paris Pamfilos