Computer Algebra: Algorithms and Applications

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Computer algebra systems are large software systems that allow scientists, engineers, mathematicians, and any other discipline that uses mathematics, to conduct mathematical calculations easily and efficiently. These systems can do much more than numerical computations. They have been programmed with large amounts of mathematical knowledge, and they can work with symbolic quantities (the high school student's x,y,z) as well as numbers. The system's mathematical knowledge allows them to perform high-level tasks, such as solving equations or simplifying complicated expressions. They can handle lengthy expressions that would take a human mathematician more than a page to write down by hand, and manipulate such large expressions without the errors that a human mathematician would inevitably introduce. Thus they allow scientists and engineers to take advantage of specialist mathematical knowledge that would otherwise not be available to them. Instead of the researcher hunting through the library or internet to learn how to solve some difficult equation, and then having to spend time applying the discovered knowledge to the researcher's particular problem, the algebra system user can refer the problem to the system and benefit from the expert knowledge of generations of developer mathematicians and computer scientists. Scientific investigations can gain greatly in mathematical level and productivity.One such computer system is Canadian. It is named, appropriately, Maple. The results of research supported in this proposal will be openly published, but the results will relevant to improving Maple. This project will increase the power of existing computer algebra systems by adding to their mathematical knowledge base. The mathematical functions known to the systems will be investigated and new properties discovered that will help systems solve users' problems. Of particular importance to the systems and their users are properties connected with complex numbers. Complex numbers are those built using the square root of negative one. Working with functions of complex numbers, and simplifying the mathematical expressions which arise, is a long standing difficulty, which is still not adequately solved. This project aims to strengthen the abilities of the systems in this area.