The goal of the course is to understand the underlying principles of
algorithms typically used in computer algebra systems for computing closed
forms of integrals, or for solving differential equations.
The topics include: Differential fields, integration of rational functions,
the Risch algorithm, polynomial and rational solutions of ODEs,
hyperexponential solutions, series solutions via Frobenius method and
Newton polygon, algorithms for algebraic and holonomic functions.
Bachelor or Masters theses topics which are based on the material of this course
- Lecturer: Manuel Kauers
- If you are regular JKU student, please register to this course via
- Exam/credits mode: will be discussed in the first meeting
- Time/Place, 16:15--17:45, HS14
There is no lecture on June 6. Instead, we will prolong by
one week and have an extra meeting on July 3.
- First meeting: March 06
Participants are expected to be acquainted with the basic notions from analysis
(differentiation, integration, power series expansions, etc.) and
linear algebra (finite dimensional vector spaces, solving linear systems of equations, etc.).
It might be an advantage, although it is not a formal requirement, to do the courses Computeralgebra
and/or Analytische Kombinatorik before this course.
- Lecture notes:
- Motivation, Differential Rings, and the Problem(s) of Symbolic Integration
- In-Field Integration of Rational Functions: Hermite Reduction
- Elementary Integration of Rational Functions: Rothstein-Trager Resultant
- Liouville's Theorem
- Risch's Algorithm, part I: Hermite Reduction and Rothstein-Trager Resultant
- Risch's Algorithm, part II: The Polynomial Part
- General remarks about ODEs; polynomial solutions
- rational solutions; generalized series solutions I (Frobenius method)
- generalized series solutions II (Newton polygon); hyperexponential solutions
- Picard-Vessiot rings
- Differential Galois groups
- Closure Properties for D-finite functions
- Several variables; definite integration