Re: Another damn simplifying problem: ArcTan

*To*: mathgroup at smc.vnet.net*Subject*: [mg60126] Re: Another damn simplifying problem: ArcTan*From*: David Bailey <dave at Remove_Thisdbailey.co.uk>*Date*: Sat, 3 Sep 2005 02:06:20 -0400 (EDT)*References*: <dek7hq$a2t$1@smc.vnet.net> <demm3s$rd1$1@smc.vnet.net> <dep7cc$euf$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Mathieu McPhie wrote: > Hi all again, > > By drawing a little diagram I have cunningly devised the following > arctan indentities, which I think M. should know, you know? > > 1. ArcTan[x,y] + ArcTan[x,-y] = 0, for all real x and y > 2. ArcTan[x,y] + ArcTan[-x,y] = pi, for all real x and y > 0 > 3. ArcTan[x,y] + ArcTan[-x,y] = -pi, for all real x and y < 0 > > I can get M to Simplify the 1st expression, but only for x > 0, not the > general result, i.e. > > Simplify[ArcTan[x,y]+ArcTan[x,-y],x>0] = 0 > > I can get Ma to reproduce the first general by the following complicated > expression > > Simplify[Factor[TrigToExp[ArcTan[x, y] + ArcTan[x,-y]]] > /. Log[x_] + Log[y_] -> Log[x y]] = 0 > > But using the same proceedure with the 2nd/3rd expression yields the > answer pi, regardless of the sign of y. > > Simplify[Factor[TrigToExp[ArcTan[x, y] + ArcTan[-x,y]]] > /. Log[x_] + Log[y_] -> Log[x y]] = pi > > WTH? Anyone got ideas here? > > Cheers, Mat > > Mathieu McPhie wrote: > >>Sorry, I got my x's and y's mixed up, and so I was more curious about >>why "M" can't simplify the following: >> >>Simplify[ArcTan[x,-y]+ArcTan[x,y]] >> >>This can be simplified by choosing x > 0, and not if x < 0. Which >>according to the range of the ArcTan function should also evaluate to 0. >> >>Cheers, Mat > > Hello, Simplify will never supply everything that might be needed, but why not write your own function that looks for expressions like ArcTan[x,y]+ArcTan[x,-y] and evaluates them. Of course, your own function can even make simplifications that are not valid in all cases (or ever!). David Bailey http://www.dbaileyconsultancy.co.uk