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Re: Another damn simplifying problem: ArcTan

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59974] Re: Another damn simplifying problem: ArcTan
  • From: Mathieu McPhie <m.mcphie at fz-juelich.de>
  • Date: Sat, 27 Aug 2005 04:11:04 -0400 (EDT)
  • References: <dek7hq$a2t$1@smc.vnet.net> <demm3s$rd1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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


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