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