Qualitative analysis of mathematical expressions using computer symbolic mathematics

Increasingly powerful computers and simplification algorithms permit us to obtain answers for increasingly complex computer-algebra problems. Consequently, we will continue to get results which often are incomprehensibly lengthy and complicated. However, a user of computer-algebra systems need not abandon hope when faced with such results. Often the user is interested in qualitative properties of a result rather than details of an analytical representation of the result. For example, is the result real? bounded? even? continuous? positive? monotonic? differentiable? or convex? Where, if any, are the singularities, zeros, and extrema? What are their orders? What is the local behavior in the neighborhood of these notable features, as exhibited perhaps by series expansions? Are there simple asymptotic representations as certain variables approach infinity? This paper describes a program which automatically analyzes expressions for some of these properties. The user may enquire about a specific property, such as monotonicity, or he may simply invoke a single function which attempts to determine all of the properties addressed by the collection of more specific functions. The specific functions are appropriate when a user knows which properties are important for his application, but frequently he is ignorant of the most decisive questions or ignorant of specific available functions which automatically investigate the desired properties. The collective qualitative analysis function is intended as a sort of panic button, which hopefully will provide some pleasantly surprising results that serve as a point of departure for further analysis. This function is a tool for deciphering unwieldy expressions that otherwise defy understanding. Many of the above qualitative properties have numerous testable characterizations, and only a few have been explored here. However, the results of this initial effort indicate that qualitative analysis programs are a promising means of extending the utility of computer algebra.