Monday, November 14, 2016


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:

Puiseux expansions around singular points and their radii of convergence:

Iterated exponential functions:

Folded polynomial functions:


  1. 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.

    1. Noted. Thanks. I restricted the definition to the type of functions covered and given by (1) in the home page.

  2. Dear dominic,
    I 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


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