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A website about algebraic functions

This web site is about a particular class of multi-valued functions and is currently under construction. When completed, it will give readers a practical guide to understanding what algebraic functions are, how to plot, illustrate and analyze contour integrations over them, explain how to compute power expansions of these functions, explain, implement and code the Newton Polygon algorithm, and show how the entire foundation of Complex Analysis for single valued functions is applicable to algebraic functions. For example, how does one apply Laurent's Theorem to algebraic functions? How can a (convergent) power expansion of an algebraic function extend across multiple singular points? The following sections answer these questions. The reader is advised to read the sections in sequential order as each section builds and makes frequent reference to previous sections.

The software used in this web site is Mathematica.

Choose pages in menu above or click on the list below:

• 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

The Mathematica Code page will contain the complete code to implement the Newton Polygon Application and other concepts reviewed.