Risch Integration

Spring 2006
University of Maryland, College Park

non-credit course


In 1968, Robert Risch, using abstract algebra, solved a centuries-old problem in mathematics - given an elementary function f(x), can its integral be written as another elementary function F(x)? Risch's technique either finds the integral or proves that it doesn't exist. Refined and improved, it has found its way into computer programs like MAXIMA, Mathematica, AXIOM, and MAPLE, to name a few. Unfortunately, calculus is still taught the "old way" - after presenting a hodgepodge of techniques (integration by parts, trigonometric substitutions, change of variables) the student is basically told, "now if you really want to solve an integral, you use a computer." This class teaches how the computer does it, not through massive approximation or exhaustively checking thousands of integrals, but in a framework that may ultimately define how computers will be used in twenty-first century mathematics - hard theorems, tight proofs, solid math.





We have no textbook as such, partly because I know of no single text that covers all of this material. By the end of the class, we'll be into various research papers. These are my main references:

Geddes, Czapor, Labahn, "Algorithms for Computer Algebra" (1992). ISBN 0-7923-9259-0

Bronstein, "Symbolic Integration I: Transcendental Functions", 2nd ed (2005). ISBN 3-540-21493-3 Trager, "Integration of Algebraic Functions", Ph.D. thesis (1984). Bronstein, "Integration of Elementary Functions", J Symbolic Computation (1990) 9, 117-173.

Lecture Recordings:

Lectures will be conducted using a tablet computer and LCD projector to do all the math. The computer's screen and classroom audio will be recorded, converted to MPEG-4 (DivX) and posted, typically within 24 hours. All recorded material is public domain; no rights reserved; copying and redistribution freely permitted. DivX players are available from divx.com; video iPods may also be capable of playing these videos.

Links to the MPEG-4 AVI files are slotted into the (continually revised) syllabus as they become available.


1 week - Introduction

2-3 weeks - Algebra 1-2 weeks - Algebraic structure of Risch integration 1 week - Differentiation 1 week - Integration in a differential field Integration by factoring the denominator and partial fractions. Integration with Rothstein/Trager resultants. Hermite reduction. Problems created by multi-valued nature of the complex logarithm. 2-3 weeks - Integration in logarithmic extensions 3-4 weeks - Integration in exponential extensions 3-4 weeks - Integration in algebraic extensions (Trager's method) any remaining time - Integration in algebraic extensions (Bronstein's method)

Thurs May 11 - Last class