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Re: ArcSin[Sin[x]]

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg2048] Re: [mg2008] ArcSin[Sin[x]]
• From: Richard Mercer <richard at seuss.math.wright.edu>
• Date: Sat, 16 Sep 1995 01:43:28 -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.....

Jack,
Thanks for your comments.
I am not at all opposed to simplifying ArcSin[Sin[x]] to x, just to doing it  
globally and/or in ignorance of the mathematical facts! I suspect you  
understood me on this.

In that respect it is fine to have some special command (such as PowerExpand)  
give this result. If it were my choice I wouldn't use an error message. As you  
point out, much that PowerExpand already does is only true with some  
restrictions. 


Series can be expected to give results only applicable to a restricted  
interval, since this is a typical situation for power series.

On a related subject from John Burnette,
>  I didn't understand the reference to Log[Exp[x]], it
>  doesn't seem to be a parallel situation. In what cases
>  is Log[Exp[x]]<>x ?

Log[Exp[x]] is equal to x for all real numbers x.
For complex numbers this only true if the imaginary part lies between -Pi and  
+Pi (as implemented in Mathematica).
Due to the periodic nature of imaginary (and hence complex) exponentials
(   exp(i x) = cos(x) + i sin(x)   )
the Exp function is not invertible over all complex numbers. 

I don't know if the original poster of the Log[Exp[x]] question was concerned  
with complex numbers or not.

Richard Mercer