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MathGroup Archive 1995

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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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