D-Finite Functions
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Beschreibung
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.
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
Autor*in
Manuel Kauers
Themen in »D-Finite Functions«
D-finite functions differentiably finite functions holonomic functions discrete mathematics linear recurrences differential equations symbolic computation computer algebra special functions symbolic summation and integration Ore algebras
Stimmen zu »D-Finite Functions«
“The aim of this book is to provide a solid foundation for researchers who wish to enter the field of D-finite functions. It offers a comprehensive and self-contained introduction to the theory, computation, and applications of D-finite functions, making it particularly valuable for researchers and graduate students. The book contains many illustrative examples to facilitate the reading of the book.” (Amir Hashemi, zbMATH 07756344, 2024)
“The book is written in a precise style with attention to detail. The provided algorithms are in symbolic code, but can be easily (relatively) translated to programs. The exercises are well designed and informative. The book is an excellent source of information and will work best for graduate-level math education, though it may also motivate researchers to apply the theory in their own fields.” (Alexander Tzanov, Computing Reviews, March 27, 2024)
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