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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
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