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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
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