Re: A very unexpected result for a Taylor Series
- To: mathgroup at smc.vnet.net
- Subject: [mg9827] Re: A very unexpected result for a Taylor Series
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Fri, 28 Nov 1997 05:35:40 -0500
- Organization: University of Western Australia
- Sender: owner-wri-mathgroup at wolfram.com
Wretch wrote:
> Now, suppose we call the expanded function F[q]. I typed in
>
> blah = Series[F[q],{q,0,3}] ,
>
> and it returned fractional powers of q in the alleged Taylor series. The
> output was
>
> a q^{1/3} + b q + c q^{5/3} + d q^{7/3} + O(q^{10/3}) ,
>
> where a,b,c,d are ugly looking constants.
>
> So, it truncated before it got past powers higher than 3, but what's
> with the appearance of these fractional powers? Any advice?
As the Mathematica book says
- Series can construct standard Taylor series, as well as certain
expansions involving negative powers, fractional powers and logarithms.
Here is a simple example that will lead to fractional powers of the type
you encountered:
In[1]:= Series[(q + q^2)^(1/3), {q, 0, 3}]
Out[1]=
4/3 7/3
1/3 q q 10/3
q + ---- - ---- + O[q]
3 9
It is clear that (q + q^2)^(1/3) does _not_ possess a standard Taylor
series about q==0. However, the above result is still very useful.
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.pd.uwa.edu.au/~paul
God IS a weakly left-handed dice player
____________________________________________________________________