Re: Re: Another damn simplifying problem: ArcTan
- To: mathgroup at smc.vnet.net
- Subject: [mg60003] Re: [mg59982] Re: [mg59910] Another damn simplifying problem: ArcTan
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sun, 28 Aug 2005 03:07:38 -0400 (EDT)
- References: <200508251033.GAA10111@smc.vnet.net> <200508270811.EAA15239@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 27 Aug 2005, at 09:11, Andrzej Kozlowski wrote: > On 25 Aug 2005, at 11:33, Mathieu McPhie wrote: > > >> Can someone please explain this to me: (M v4.something) >> >> In:= Simplify[ArcTan[-x]+ArcTan[x]] >> Out= 0 >> >> In:= Simplify[ArcTan[-x,1]+ArcTan[x,1]] >> Out= ArcTan[-x,1]+ArcTan[x,1] >> >> Note, I want something more complicated than this obviously. Actually >> want something like >> >> Simplify[ArcTan[-x,y]+ArcTan[x,y]] >> >> but above is the easiest example of this infuriating programs >> problem. >> >> Cheers, Mathieu >> >> >> > > > In Mathematica 5.1 the situation does not look much better. > > > First however, note that > > > Simplify[ArcTan[x,-y]+ArcTan[x,y],{x>0}] > > 0 > > Unfortunatley > > Simplify[ArcTan[x,-y]+ArcTan[x,y],{x<0}] returns > > ArcTan[x,-y]+ArcTan[x,y] instead of 0. > > Adding assumptions on y does not help. > > Your formula > > ArcTan[x,-y]+ArcTan[x,y] > > is not surprisingly harder for Mathematica to simplify since we have: > > > ArcTan[-x, y] + ArcTan[x, y] /. {x -> 1, y -> -1} > > > -Pi > > > ArcTan[-x, y] + ArcTan[x, y] /. {x -> 1, y -> 1} > > Pi > > etc. > > > > > > Still Mathematica ought to manage these cases with assumptions on x > and y and it doesn't. In fact, for x>0 one can get the answer isn > this way: > > > Simplify[ComplexExpand[ArcTan[x, -y] + ArcTan[x, y],TargetFunctions-> > {Re,Im}], {x > 0}] > > 0 > > > > > Andrzej Kozlowski > > > This last line was of course a mistake, I did not notice that I had ArcTan[x,-y]+ + ArcTan[x, y] ( which already works without ComplexExpand ) instead of ArcTan[-x, y] + ArcTan[x, y]. However, we actually can get the "correct answer" as follows: PowerExpand[FullSimplify[TrigToExp[ArcTan[-x, y] + ArcTan[x, y]], {x > 0, y > 0}]] Pi However, our excitement is somewhat damped by: PowerExpand[FullSimplify[TrigToExp[ArcTan[-x, y] + ArcTan[x, y]], {x > 0, y < 0}]] Pi which is only correct "up to sign" (the answer is -Pi). Of course using PowerExpand alone "proves" little, but combined with numerical verification such as ArcTan[-x, y] + ArcTan[x, y] /. {x -> Rationalize[Random[], 0], y -> Rationalize[Random[], 0]} Pi ArcTan[-x, y] + ArcTan[x, y] /. {x -> Rationalize[Random[], 0], y -> -Rationalize[Random[], 0]} -Pi it could be reasonably treated as a "probabilistic proof". Andrzej Kozlowski
- References:
- Another damn simplifying problem: ArcTan
- From: Mathieu McPhie <m.mcphie@fz-juelich.de>
- Re: Another damn simplifying problem: ArcTan
- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
- Another damn simplifying problem: ArcTan