Volume 6 - Number 4 - Viewing the Roots of Polynomial Functions in Complex Variable: The Use of GeoGebra and the CAS Maple

Oct 15, 2026

Uploaded by Putri

Volume 6, Number 4 (2013) presents a discussion of the Fundamental Theorem of Calculus within Complex Analysis, focused on analyzing polynomial equation roots. It connects the theorem with Rouché’s theorem, the winding number, and the Argument Principle, emphasizing root-finding consequences in the complex plane. The work proposes a complementary workflow using GeoGebra for dynamic visualization and CAS Maple for complex algebraic computation and 3D object visualization.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

AccessFree to download

Introduction

Fundamental theorems and barriers in Complex Analysis

Rouché’s theorem, Argument Principle, and winding number

Heuristic determination and visualization of winding numbers

Volume 6, Number 4, 2013 VIEWING THE ROOTS OF POLYNOMIAL FUNCTIONS IN COMPLEX VARIABLE: THE USE OF GEOGEBRA AND THE CAS MAPLE Francisco Regis Vieira Alves Abstract: Admittedly, the Fundamental Theorem of Calculus – TFA holds an important role in the Complex Analysis - CA, as well as in other mathematical branches. In this article, we bring a discussion about the TFA, the Rouché´s theorem and the winding number with the intention to analyze the roots of a polynomial equation. We propose also a description for a method that involves a complementary perspective for the use of two mathematical softwares. Thus, with the intention of cheking the TFA and other mathematical properties related with the polynomial equation roots, we enphazise the visualization, an exploration with the Geogebra and the computer algebric system - CAS Maple. This CAS permits execute complex algebric operations and the visualization of tridimensional objects. On the other hand, the software Geogebra makes possible the dynamic mathematical exploration of the sciencific concepts in the teaching context of CA. Key words: Fundamental Theorem of Algebra, Visualization, Software Geogebra, CAS Maple, Winding number. 1. Introduction In academic locus, we find several scientifics topics that can promote some barriers to the teaching and mathematical learning (Alves, 2012) . In particular, we highlight in this paper some topics in the Complex Analysis (Atiyah, 2002, p. 2) . Indeed, we observe few papers about the teaching of Complex Analysis. In Brazil, we constate some works about the teaching/learning of CA. With a great attention, we realize the abstract character of several concepts in this particular mathematical branch. Moreover, there are inumerous classical theorems and concepts that require a deep conceptual knowledged. Nowadays, we can not deny the possibilities and advances in technology. Thus, in a specic way, when we turn to the teaching in Mathematics, we constate softwares that allows to exploitation of certains abstract contents (Alves, 2013a, 2013b) . In this article, we take into account a complementary character ofto use the system Geogebra and the CAS Maple in the academic teaching context. Originating in these two arguments of the previous paragraphs, we put in evidence the perspective advocated by Needham (2000) . He defends, describes and suggests an approach for to CA supported by the visualization and our perception about the graphic-geometric properties. In fact, in your book intituled “Visual Complex Analysis”, we constate an approach that promotes a heuristic interpretation for the: power series of fonctions; the notion of complex derivative f '(z) ; theory of integration; behavior of the graphics of the function in the complex variable, theory of residue, etc.. Here, we express a particular interest in a classical theorem that provides implications in various branches of Mathematics. Indeed, the Fundamental Theorem of Algebra – TFA allows to conclude a strong propertie about the corp of complex numbers 􀀀 . On the other hand, we cannot discuss it without mentioning the other two elements. The first is just the theorem that implies that one of possible statements and the proof for the TFA (Bottazzini, 1986) . We talk about the Rouché´s theorem. The second element is a relevant topological argument called for “Argument Principle”. We will start to discuss some properties and barriers related to the FTA. But, we are mentioning other fundamental notion in the CA. In fact, Needham (2000, p. 338) coments that “as the name suggests, the winding number ν [L, O] of a closed loop L about the origin O is simply the net number of revolutions of the direction of z as it traces out L once in its given sense.” This author seeks to transmit his ideas not only by the formalism and the prove process related to the theorems. With a heuristic intention, Needham (2000, p. 338) shows the figures below. He describes a visual and perceptual pr

FAQs

What core theorems are used to study polynomial roots in complex variable?

The text centers on the Fundamental Theorem of Calculus/Algebra themes and connects them to Rouché’s theorem and topological arguments including the Argument Principle.

How do GeoGebra and CAS Maple complement each other in the proposed method?

GeoGebra supports dynamic, visual exploration in teaching contexts of complex analysis, while CAS Maple performs complex algebraic operations and provides visualization such as 3D objects.

How is the winding number intended to be determined visually?

The document describes a heuristic procedure: starting from a random point, tracing the loop and counting revolutions after connecting the origin to a moving finger, adding or subtracting based on counterclockwise or clockwise motion.

Oct 15, 2026

Uploaded by Putri