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Re: Series of a Root object
*To*: mathgroup at smc.vnet.net
*Subject*: [mg14296] Re: Series of a Root object
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Tue, 13 Oct 1998 01:21:14 -0400
*Organization*: University of Western Australia
*References*: <6vf5ol$dht@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
RENZONI_FERRUCCIO wrote:
> I would like to know why the following program, meant to expand the Root
> object r in serie of g and s up to the second order, fails to give an
> answer.
I'm not completely sure but I think the problem is that the there does
not exist a Taylor expansion for this particular root. To give you a
trivial example (which is actually closely related to your specific
example), consider the following:
In[1]:= (roots = Solve[16 g^2 - 8 g + 16 x^2 + 8 x + 1 == 0, x]) Out[1]=
2
-1 - 2 Sqrt[2] Sqrt[g - 2 g ] {{x ->
-----------------------------},
4
2
-1 + 2 Sqrt[2] Sqrt[g - 2 g ]
{x -> -----------------------------}}
4
If you now expand these roots into a series you obtain
In[2]:= (x/.roots) + O[g]^2
Out[2]=
3/2
1 Sqrt[g] g 2
{-(-) - ------- + ------- + O[g] ,
4 Sqrt[2] Sqrt[2]
3/2
1 Sqrt[g] g 2
-(-) + ------- - ------- + O[g] }
4 Sqrt[2] Sqrt[2]
It is clear that this series is NOT analytic (since it involves Sqrt[g])
and no Taylor series exists.
Note that you can perhaps see more clearly what is happening using Plot:
In[3]:= Plot[Evaluate[x /. roots], {g, -1, 1},
PlotStyle -> Table[Hue[i], {i, 0, 1, 1/Length[roots]}]];
You will see why no Taylor series exists at g=0.
For your root,
> r = Root[-8*b*g^4*s^2 + 8*b^2*g^4*s^2 - 4*g^4*#1 + 8*b*g^4*#1 +
> 4*b^3*g^2*s^2*#1 - 16*g^4*s^2*#1 + 24*b*g^4*s^2*#1 + 16*b*g^4*s^4*#1 -
> 2*b*g^2*#1^2 + 6*b^2*g^2*#1^2 + 16*g^4*#1^2 + 20*b^2*g^2*s^2*#1^2 +
> 32*g^4*s^2*#1^2 + 16*g^4*s^4*#1^2 + b^3*#1^3 - 4*g^2*#1^3 +
> 22*b*g^2*#1^3 + 36*b*g^2*s^2*#1^3 + 5*b^2*#1^4 + 20*g^2*#1^4 +
> 20*g^2*s^2*#1^4 + 8*b*#1^5 + 4*#1^6 & , 3];
if you extract the original polynomial,
poly = (-8*b*g^4*s^2 + 8*b^2*g^4*s^2 - 4*g^4*#1 + 8*b*g^4*#1 +
4*b^3*g^2*s^2*#1 - 16*g^4*s^2*#1 + 24*b*g^4*s^2*#1 +
16*b*g^4*s^4*#1 -
2*b*g^2*#1^2 + 6*b^2*g^2*#1^2 + 16*g^4*#1^2 + 20*b^2*g^2*s^2*#1^2 +
32*g^4*s^2*#1^2 + 16*g^4*s^4*#1^2 + b^3*#1^3 - 4*g^2*#1^3 +
22*b*g^2*#1^3 + 36*b*g^2*s^2*#1^3 + 5*b^2*#1^4 + 20*g^2*#1^4 +
20*g^2*s^2*#1^4 + 8*b*#1^5 + 4*#1^6 &)[x];
and then visualize the roots for specific b, s, and g, you may be able
to more clearly see what is going on.
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia Nedlands WA 6907
mailto:paul at physics.uwa.edu.au AUSTRALIA
http://www.physics.uwa.edu.au/~paul
God IS a weakly left-handed dice player
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