Multiple-valued complex functions and computer algebra

ABSTRACT

I recently taught a course on complex analysis. That forced me to think more carefully about branches. Being interested in computer algebra, it was only natural that I wanted to see how such programs dealt with these problems. I was also inspired by a paper by Stoutemyer ([3]).While programs like Derive, Maple, Mathematica and Reduce are very powerful, they also have their fair share of problems. In particular, branches are somewhat of an Achilles' heel for them. As is well-known, the complex logarithm function is properly defined as a multiple-valued function. And since the general power and exponential functions are defined in terms of the logarithm function, they are also multiple valued. But for actual computations, we need to make them single valued, which we do by choosing a branch. In Section 2, we will consider some transformation rules for branches of multiple-valued complex functions in painstaking detail.The purpose of this short article is not to do a comprehensive comparative study of different computer algebra system. (For an attempt at that, see [4].) My goal is simply to make the readers aware of some of the problems, and to encourage the readers to sit down and experiment with their favourite programs.I would like to thank Willi-Hans Steeb and Michael Wester for helpful comments.

REFERENCES

• 1: [1] Thomas Clausen, Aufgabe 53, J. Reine Angew. Math. 2 (1827), 286-287.
• 2: [2] Reinhold Remmert, Theory of Complex Functions, Graduate Texts in Math., vol. 122, Springer-Verlag, 1991.
• 3: [3] David R. Stoutemyer, Crimes and misdemeanors in the computer algebra trade, Notices Amer. Math. Soc. 38 (1991), 778-785.
• 4: [4] Michael Wester, A review of CAS mathematical capabilities, Computer Algebra Nederland 13 (1994) 41-48.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

General Terms:
Theory, Verification