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 are available.

- Lecturer: Manuel Kauers
- If you are regular JKU student, please register to this course via KUSSS
- 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
- Prerequisits: 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