# Computer Algebra, MAS5731, Syllabus, Fall 2000.

Location: MCH 107

The course documents and assignments

Time: Wednesday and Friday 12:30 - 1:45

Instructor: Dr. Mark VanHoeij

• e-mail: hoeij@math.fsu.edu
• URL: http://www.math.fsu.edu/~hoeij
• Office: 105B LOV. Note: this is very near where the construction is going on. When the noise gets too much (when they are drilling), talking in my office could become impossible. In such situations you may find me in the Dirac library instead of in my office, even during office hours. I apologize for the inconvenience.
• Office hours: Monday, Tuesday, Thursday. 10 - 11. You can also visit my office during other hours, but send me an e-mail first to make an appointment to make sure I'll be in my office.

You do not need a text book for this course because all documents will be placed on the web. Symbolic integration will be a large part of this course. If you want a text book on symbolic integration then take Symbolic Integration I. This is an excellent book, it is very clear, and it is written by someone who played an essential role in the development of symbolic integration. If you would like to learn other topics in computer algebra then I highly recommend: Modern Computer Algebra, which is a pleasure to read, contains all the right topics, it is the best general book to study computer algebra.

Contents Computer Algebra MAS-5731 in Fall 2000.
• Elementary Integration.

See the help page ?int in Maple for some examples. How can Maple compute such integrals? We will focus on the problem of elementary integration. A function will be called an elementary function if it can be built up with the following:
• Rational functions in x with coefficients in the complex numbers C.
• The functions exp and log.
• Additions, multiplications, divisions, and compositions of functions.
• algebraic extensions
So for example the following functions are elementary functions (note that sin(x) and cos(x) can be expressed in terms of the exponential function):

f := x/(x^2+1); g := sin(x); h := exp(cos(x)+x^2)/(ln(x)^2+1)^2;

Now the problem of elementary integration is:

Given an elementary function f, does there exist an elementary function F such that F'=f? If so, how to find F? Although this problem looks analytical, the solution to this problem is purely algebraic in nature.

• Linear Differential Equations.

ode := (x^3+x)*diff(diff(y(x),x),x)-diff(y(x),x)+y(x)*x^3 = 0;

dsolve(ode); # Maple's solver for differential equations

How does Maple compute something like this?

It turns out that many ideas used in elementary integration can also be used in linear differential equations. We will study some of the methods. Again, the algorithms we will study will be algebraic in nature instead of analytical.

Grading: There will be two tests during the semester and a final test. Each of these three tests will account for 20% of the final grade. The remaining 40% of the grade will be determined by the turn in assignments, which will all (or almost all) be computations or algorithms you have to write in Maple. So it is essential to learn Maple, and we will spend time on that during the course.

To have access to Maple you need to have an account on one of the Unix machines. You can also buy Maple in the book store for \$129. Most of the Unix machines have Maple release 5, but the new version (in the book store) is Maple 6. So I will write the Maple documents in such a way that they will work in Maple 5 as well as Maple 6 (or I will point out what needs to be changed in the document between the two versions).