Algebraic functions, iterated exponentials and polynomial systems

A website about algebraic functions and iterated exponential and polynomial systems

$$ \newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} \newcommand{\wvalx}{\underbrace{z^{\lambda_4}(c_4+w_5)}_{w_4}} \newcommand{wvalxx}{\underbrace{z^{\lambda_3}(c_3+\wvalx)}_{w_3}} \newcommand{wvalxxx}{\underbrace{z^{\lambda_2}\{c_2+\wvalxx\}}_{w_2}} \newcommand{wvalxxxx}{z^{\lambda_1}\big(c_1+\wvalxxx\big)} $$

Upcoming Two‑Volume Book Release (Early August 2026)

Over the past several years, the posts on this blog have explored various aspects of algebraic functions, often in a concise or introductory way. I am pleased to share that a much more detailed and fully developed undergraduate introduction of this material will soon be available in the form of a two‑volume text titled Algebraic Functions: A Computational Introduction Using Mathematica.

The following is an excerpt from the preface of Volume 1:

Multi-valued functions can often be a formidable and sometimes discouraging topic, yet these functions lie at the heart of some of the most beautiful ideas in mathematics. This two-volume set offers an accessible and innovative path into the subject through the study of algebraic functions and is written for readers with a background in complex variables, a basic knowledge of Mathematica, and some exposure solving initial value problems numerically, a task easily done in Mathematica. The first volume consists of Chapters 1-6 and introduces the branching geometry of algebraic functions, the Newton polygon method for computing branch power expansions centered at a point (referred to in this book as central expansions), a method for determining the radii of convergence of central expansions, accuracy profiles of central expansions, and a technique for computing power expansions of annular branches by applying Laurent's theorem to algebraic functions. Volume 2 continues this analytic and geometric framework with Chapters 7 through 12 by computing the radii of convergence and accuracy profiles of annular branches and developing several analytic properties of annular power expansions. Chapter 10 introduces Newton polygon types for estimating the genus of an algebraic function by inspection, and Chapter 11 presents a method for computing the exact genus from the function's global ramified geometry. The final chapter presents a detailed discussion of the principle of analytic continuation and its application to the Beta, Gamma, and Zeta functions, including the extension of the Beta function to complex exponents resulting in multi-valued functions with infinitely ramified coverings.

The exposition begins with elementary ideas, clearly explained, and gradually develops more sophisticated topics in a way that remains approachable to the motivated reader. By the end of the final chapter, the reader should feel comfortable and confident working with the analytic geometry of intricate contour integrals over multi-valued functions. No background in advanced mathematics is needed to understand the topics as presented in these volumes.

The printed set expands substantially on the topics introduced in this website. Each chapter includes a guided tutorial for an accompanying Mathematica notebook, providing hands‑on exploration of the concepts introduced in the text.

I am currently reviewing the final proof of Volume 2, which should arrive in mid‑July. Once both volumes are approved, I will order an initial supply of each volume. This places the expected availability of the two‑volume set in early August 2026.

A direct‑purchase option (at a reduced price compared to Amazon) will be offered here on the blog once the books are ready. More details will follow as the release date approaches.

If you would like more information or wish to express early interest in the book, you are welcome to contact me at youierns@gmail.com.