Methods for finding Zeros in Polynomials
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About the book
Description
In this book you find the basic mathematics that is needed by engineers and university students . The author will help you to understand the meaning and function of mathematical concepts. The best way to learn it, is by doing it, the exercises in this book will help you do just that.
Polynomials are the first class of functions that the student meets. Therefore, one may think that they are easy to handle. They are not in general! Topics as e.g. finding roots in a polynomial and the winding number are illustrated. Some of the topics only require an elementary knowledge of Calculus in one variable. Others rely heavily on Complex Functions Theory.
Content
Introduction
1 Complex polynomials in general
1.1 Polynomials in one variable
1.2 Transformations of real polynomials
1.2.1 Translations
1.2.2 Similarities
1.2.3 Reflection in 0
1.2.4 Inversion
1.3 The fundamental theorem of algebra
1.4 Vieti’s formulæ
1.5 Rolle’s theorems
2 Some solution formulæ of roots of polynomials
2.1 The binomial equation
2.2 The equation of second degree
2.3 Rational roots
2.4 The Euclidean algorithm
2.5 Roots of multiplicity > 1
3 Position of roots of polynomials in the complex plane
3.1 Complex roots of a real polynomial
3.2 Descartes’s theorem
3.3 FourierBudan’s theorem
3.4 Sturm’s theorem
3.5 Rouch´e’s theorem
3.6 Hurwitz polynomials
4 Approximation methods
4.1 Newton’s approximation formula
4.2 Graeffe’s rootsquaring process
4.2.1 Analysis
4.2.2 Template for Graeffe’s rootsquaring process
4.2.3 Examples
5 Appendix
5.1 The binomial formula
5.2 The identity theorem for convergent power series
5.3 Taylor’s formula
5.4 Weierstraß’s approximation theorem
Index
About the Author
Leif Mejlbro was educated as a mathematician at the University of Copenhagen, where he wrote his thesis on Linear Partial Differential Operators and Distributions. Shortly after he obtained a position at the Technical University of Denmark, where he remained until his retirement in 2003. He has twice been on leave, first time one year at the Swedish Academy, Stockholm, and second time at the Copenhagen Telephone Company, now part of the Danish Telecommunication Company, in both places doing research.
At the Technical University of Denmark he has during more than three decades given lectures in such various mathematical subjects as Elementary Calculus, Complex Functions Theory, Functional Analysis, Laplace Transform, Special Functions, Probability Theory and Distribution Theory, as well as some courses where Calculus and various Engineering Sciences were merged into a bigger course, where the lecturers had to cooperate in spite of their different background. He has written textbooks to many of the above courses.
His research in Measure Theory and Complex Functions Theory is too advanced to be of interest for more than just a few specialist, so it is not mentioned here. It must, however, be admitted that the philosophy of Measure Theory has deeply in uenced his thinking also in all the other mathematical topics mentioned above.
After he retired he has been working as a consultant for engineering companies { at the latest for the Femern Belt Consortium, setting up some models for chloride penetration into concrete and giving some easy solution procedures for these models which can be applied straightforward without being an expert in Mathematics. Also, he has written a series of books on some of the topics mentioned above for the publisher Ventus/Bookboon.