MathGroup Archive 2007

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: showing an expression equal to 0

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73703] Re: [mg73719] showing an expression equal to 0
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 26 Feb 2007 06:06:10 -0500 (EST)
  • References: <200702250941.EAA08417@smc.vnet.net>

Actually, in this case you only need the observation that the  
function is complex analytic in y for R>0 in the interior of the disk  
Abs[y]<R. Now:

expr = ArcTan[y/Sqrt[R^2 - y^2]] - ArcSin[y/R]

=
Assuming[R>0,FullSimplify[D[expr,y]]]


0

and


Assuming[R>0,FullSimplify[expr/.y->0]]


0


This tells us not only that the derivative vanishes but also that all  
the higher derivatives at all the points in the interior of the disk  
vanish in the disk (and in particular at 0). In other words, the  
Taylor series at 0 is 0, which means that the function is 0 in the  
interior of the disk. So in fact this case is also easy to deal with  
by using the method "differentiate and FullSimplify".

Andrzej Kozlowski



On 25 Feb 2007, at 10:41, dimitris wrote:

> Consider
>
> In[87]:=
> expr = ArcTan[y/Sqrt[R^2 - y^2]] - ArcSin[y/R];
>
> The expression for R>0 and -R<y<R is equal to zero
>
> In[72]:=
> (Plot[Chop[Evaluate[expr /. R -> #1]], {y, -#1, #1}, Axes -> False,
> Frame -> True] & ) /@ Range[5]
>
> Based on Andrzej's resposnse on a recenet message I am able to show
> this equality for
> given R.
>
> E.g. for R=4 the following demonstrates that expr is equal to zero
>
> In[99]:=
> Simplify[expr /. R -> 4 /. y -> 2]
>
> Out[99]=
> 0
>
> In[97]:=
> D[expr /. R -> 4, y]
> Factor[%]
>
> Out[97]=
> -(1/(4*Sqrt[1 - y^2/16])) + (y^2/(16 - y^2)^(3/2) + 1/Sqrt[16 - y^2])/
> (1 + y^2/(16 - y^2))
> Out[98]=
> 0
>
> However not specifying R (but assuming R>0&&-R<y<R) I am not able to
> show that
> expr is equal to zero.
>
> Any ideas?
>
> Thanks!
>
>



  • Prev by Date: Re: PolyLog help
  • Next by Date: Re: deleting a title or subtitle in a notebook
  • Previous by thread: showing an expression equal to 0
  • Next by thread: Re: showing an expression equal to 0