MSW02.mw
Preface
Groebner Bases for Polynomial Systems
Chapter Synopsis
Canonical Forms for Polynomials with Side Relations
Univariate Side Relations
Algorithm to Transform an Expression into Canonical Form
Example 1
Example 2
Multivariate Side Relations
Maple syntax for simplification w.r.t. side relations
Example 3
Groebner Basis Preliminaries
Some Remarks about Groebner Bases
Example 4
Definition of a Groebner Basis
Definition 1
Example 5
Definition 2
Definition 3
Theorem 1
Concluding Remarks on Computing Groebner Bases
Example 6
Solving Systems of Polynomial Equations
Example 7
Example 8
Example 9
Example 10: Intersection of Surfaces
Concluding Remarks
Digression: What is a "Closed Form" Solution?
Chapter Synopsis
Example 1: Polynomial Root Finding
Example 2: Solving Non-polynomial Equations
Example 3: Functions Defined via the RootOf Construct
Symbolic Manipulation of the RootOf Construct
Final Remarks About the RootOf Construct
Symbolic Integration
Chapter Synopsis
Risch Algorithm for Elementary Functions
Example 1.1
Defining the function field
Notation for elementary functions
Examples in common notation
The same examples in exp-log notation
Liouville's Principle
Integral of a rational function
Example 1.2
Hermite Reductions for the Rational Part
Example 1.3
Rothstein-Trager Theorem for the Logarithmic Part
Example 1.4
Example 1.5
The case of transcendental elementary functions
Example 1.6
Example 1.7
Example 1.8
Integral of a logarithmic extension
Hermite Reductions for the Rational Part (log extension)
Rothstein-Trager Theorem: log extension
Example 1.9
Example 1.10
Polynomial Part (log extension)
Example 1.11
Example 1.12
Integral of an exponential extension
Hermite Reductions for the Rational Part (exp extension)
Rothstein-Trager Theorem: exp extension
Example 1.13
Polynomial Part (exp extension)
Example 1.14
Comments on algebraic and mixed extensions
Comments on non-elementary extensions
Symbolic Solution of Definite Integrals
Fundamental Theorem of Calculus: practical issues
Example 2.1
Example 2.2
Example 2.3
Example 2.4
Methods for Special Functions: Generalized hypergeometric function and Meijer G function
Example 2.5
Example 2.6
Hybrid Symbolic-Numeric Integration
Chapter Synopsis
Example 1
Example 2
Example 3: Subtracting off a singularity
Example 4: Algebraic transformation of variables
Example 5: Integrating a series term-by-term
Multidimensional Numerical Integration
Special (List) Syntax for Multiple Integrals
Example 6: Multidimensional integration
Bibliography