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 ____________________________________________________________________