Re: x-ArcSin[Sin[x]]
- To: mathgroup at smc.vnet.net
- Subject: [mg50035] Re: [mg50007] x-ArcSin[Sin[x]]
- From: DrBob <drbob at bigfoot.com>
- Date: Thu, 12 Aug 2004 05:43:58 -0400 (EDT)
- References: <200408110953.FAA04108@smc.vnet.net>
- Reply-to: drbob at bigfoot.com
- Sender: owner-wri-mathgroup at wolfram.com
>> I would like to know if anything has been changed in the Mathematica implementation of >> ArcSin[Sin[x]]. There is not, and cannot be, an implementation of ArcSin[Sin[x]]. Sin and ArcSin are separate functions, separately implemented. Moreover, Sin is periodic on the real line, and hence it cannot have a left inverse for all x. Therefore, as explained at the link, ArcSin[Sin[x]]==x is false in general. If you want it to be true for x in a certain interval, you may be able to define an appropriate branch of ArcSin FOR THAT INTERVAL. That is, it wouldn't be a function of x alone, but also a function of the interval. As x varies from 0 to 2Pi, the quantity Pi/2-x sweeps out the following interval: Pi/2 - Interval[{0, 2*Pi}] Interval[{-((3*Pi)/2), Pi/2}] Max[%] - Min[%] 2*Pi Now, on any interval of that length, Sin takes on all values between -1 and 1 TWICE. So again, on an interval like that, Sin has no left inverse. In summary: the value you give for that integral is wrong, no matter how many books agree with you. Bobby On Wed, 11 Aug 2004 05:53:03 -0400 (EDT), Paul Muller <paul.muller-at-epfl.ch at sicinfo2.epfl.ch> wrote: > Dear all, > > I know this question was already discussed in 1995 (that's what I found in Google), but > I would like to know if anything has been changed in the Mathematica implementation of > ArcSin[Sin[x]]. > > http://groups.google.ch/groups?q=Mathematica+%2BArcSin%5BSin&hl=de&lr=&ie=UTF-8&selm=DDwG4y.GrJ%40wri.com&rnum=1 > > I am calculating the magnetic field in any point of the plane of a spire (or spiral > inductor). When I apply this to the center of the spire, I get the following integral: > > Integrate[Sin[x+ArcSin[Sin[Pi/2-x]]], {x, 0, 2Pi}] > > According to textbook results for magnetic fields, the result should be 2Pi. Because of > the definition of ArcSin[Sin[x]] in Mathematica, the integral results in Pi. > Maybe I'm not careful enough when using the ArcSin function on the angles of > my geometric problem (I'm an engineer, not a mathematician), but it would > be helpful if this integral was solved "correctly". > > Any updates on this are welcome. > > Thanks in advance > > Paul Muller > > > -- DrBob at bigfoot.com www.eclecticdreams.net
- References:
- x-ArcSin[Sin[x]]
- From: Paul Muller <paul.muller-at-epfl.ch@sicinfo2.epfl.ch>
- x-ArcSin[Sin[x]]