Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of explicit expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail.

The book comes with batteries included in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix.

When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.

Offers a comprehensive introduction into theory and techniques for D-finite functions Contains many practical algorithms, carefully illustrated with detailed examples Includes hundreds of original exercises with solutions for training

Autorentext

Manuel Kauers ist als Professor für Algebra an der Johannes Kepler Universität in Linz tätig. Er befasst sich vor allem mit Computeralgebra, einem Forschungsgebiet im Grenzbereich zwischen Mathematik und Informatik. Neben zwei Fachbüchern über Computeralgebra hat er gemeinsam mit Koautoren aus aller Welt mehr als hundert wissenschaftliche Artikel publiziert und wurde mehrfach für seine Arbeit ausgezeichnet.

Inhalt
Chapter. 1. Background and fundamental concepts.- Chapter. 2. The recurrence case in one variable.- Chapter. 3. The differential case in one variable.- Chapter. 4. Operators.- Chapter. 5. Summation and integration.- Answers to Exercises.- Software.- Notations.- References.- Index.