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MathGroup Archive 2002

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Re: Re: Pb Limit ArcTan to : - Infinity

  • To: mathgroup at smc.vnet.net
  • Subject: [mg34702] Re: [mg34645] Re: [mg34624] Pb Limit ArcTan to : - Infinity
  • From: David Withoff <withoff at wolfram.com>
  • Date: Sat, 1 Jun 2002 04:29:14 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Yes, that's basically it.  The Series function does not have a Direction
option, so expansions around singular points where the expansion is
different in different directions may or may not give an expansion in
the direction that you wanted.  Expanding ArcTan[x] around infinity
is like expanding ArcTan[1/x] around zero, where the expansion obviously
depends on direction (see for example Plot[ArcTan[1/x], {x, -1, 1}]).

If a general computer algorithm is ever developed for controlling the
direction of expansions around singularities this would certainly
be a useful addition to Mathematica, precisely to handle cases
like this.

> i think the answer is that Series does not expand around -Infinity, only 
> about Infinity. Presumably this is because the expansions are meant to 
> be interpreted as given in the complex plane. You can check that using 
> -Infinity and Infinity gives the same series (at least in all the cases 
> I have checked). However, you can always use:
> 
> In[16]:=
> Normal[Series [ ArcTan [ -x ] , { x ,  Infinity , 2 }]]
> 
> Out[16]=
>    Pi    1
> -(--) + -
>    2     x
> 
> Andrzej Kozlowski
> Toyama International University
> JAPAN
> http://platon.c.u-tokyo.ac.jp/andrzej/
> 
> On Thursday, May 30, 2002, at 03:55  PM, Didier Losfeld wrote:
> 
> > Bonjour ,
> >
> > I do not understand answers of Mathematica :
> >
> > question 1 : Series [ ArcTan [ x ] , { x , - Infinity , 2 } ]
> >
> > answer : Pi / 2 - 1 / x + O ( 1 / x ) ^ 3
> >
> > question 2 : Limit [ ArcTan [ x ] , x -> - Infinity ]
> >
> > answer : - Pi / 2
> >
> > The first answer is False , the second answer is correct.
> >
> > Many thanks in advance.
> >
> > Didier Losfeld Professor of mathematics ( France )


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