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

```

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