ArcSin[Sin[x]]

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg2008] ArcSin[Sin[x]]*From*: Jack Goldberg <jackgold at math.lsa.umich.edu>*Date*: Thu, 14 Sep 1995 23:27:36 -0400

I am keenly interested in how one should handle ArcSin[Sin[x]]. In fact, I have twice posted questions related to this issue. So, I was delighted that Richard Mercer decided to throw his hat in the ring on this issue. I believe it should be discussed openly and constructively. Richard has presented one side. I have decided to take a few minutes to present the other. The main argument against having Mma simplify ArcSin[Sin[x]] to x is that it is, in one word, wrong! But wait. Isn't it also wrong to simplify Sqrt[x^2] to x? and for the same reason? Yet PowerExpand does that latter and not the former. I argue then, that if you reject ArcSin[Sin[x] = x because it is correct only for a limit selection of x then you must also reject PowerExpand. Richard points out that the plot of ArcSin[Sin[x]] is graphic evidence of the danger of reducing this expression to x. Agreed. But consider this: Series[ArcSin[Sin[x]],{x,0,9}] returns x + O[x^10] Mma seems to take sides here! I hope to offer a compromise. Suppose PowerExpand came with the option to simplify expression such as ArcSin[Sin[x]] and ArcTanh[Tanh[x]] to x WITH A WARNING. Then those with the ability to use this facility wisely benefit from it and those who are not, refrain from its use. Except for the warning message, WRI has apparantly opted for this compromise when they offered PowerExpand as a tool which simplifies both Sqrt[x^2] and Log[Exp[x]]. So why not ArcSin[Sin[x]] and its sisters? I think that our difference of opinion stems from the fact that I believe simplification tools are exploratory and need not be valid in all cases. After the simplification is made, one looks at the result and then argues about its range of applicability, whether tis unique and so forth. Why deny us a tool because it is dangerous in the hands of the inexperienced? Penicillin has a warning label, so should PowerExpand. P.S. Glad your back Richard. The place was going to the dogs without you. Jack