PowerExpand etc.

*To*: mathgroup at christensen.cybernetics.net*Subject*: PowerExpand etc.*From*: Jack Goldberg <Jack.Goldberg at math.lsa.umich.edu>*Date*: Fri, 21 Oct 1994 13:03:18 -0400 (EDT)

As part of a previous posting, I remarked that I would like to see a generalization of one feature of PowerExpand which is it's ability to simplify Sqr[x^2] and Log[Exp[x] to x. Many of you have responded warning me of the dangers in using inverse function simplifications indiscriminantly. I take your warnings seriously not only because they are on the mark but because I am also a teacher. But this is only one side of the issue. There are good reasons for having such a simplifier, in spite of its dangers. Mma uses a warning message when PowerExpand is invoked and I presumed that anyone who designes a generalization would do so as well. I do not have an explicit instance at hand, but I can imagine Mma responding to Integrate with an answer far more complicated than necessary. (Integrate[Sqrt[x/(1-x)],x] is a reasonably good example. as I have previously remarked.) If one has a generalization of PowerExpand, say InverseSimplify, then one might apply InverseSimplify to see if the complicated integral simplifies. If it does, one can then check whether the simplification makes sense (i.e.: is well defined and has the correct derivative.) I suppose whether one wishes to use such a function depends on whether you are risk averse or not. I'd take the risk!! While I'm on this subject, I wonder if anyone knows a reference to an article which discusses the whole area of inverse functions in symbolic computing. Of course the mathemtical theory is quite well elaborated on in many fine function theory books. Are there any significant papers on the implementation of this theory for symbolic computing? I think it is the most commonly confused and controversial topic in all the news groups on CAS. Thanks to all of you for your observations and comments. They are welcome by me because they do exactly what they are intended to do; they stimulate my thinking and I hope others as well. Jack