Re: A very unexpected result for a Taylor Series
- To: mathgroup at smc.vnet.net
- Subject: [mg9803] Re: A very unexpected result for a Taylor Series
- From: Daniel Lichtblau <danl>
- Date: Fri, 28 Nov 1997 05:35:16 -0500
- Organization: Wolfram Research, Inc.
- Sender: owner-wri-mathgroup at wolfram.com
Wretch wrote:
>
> Hello all--last night I asked Mathematica3.0 to perform a Taylor series
> expansion in the usual manner. The function being tailored was a
> slightly complicated heap of fractional powers involving the variable
> being expanded, as the function arose as a root of a cubic polynomial.
> I expected there could be trouble due to the fractional powers, but I
> figured Mathematica could handle any limits involved (L'Hopital's rule,
> etc.).
>
> 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?
>
> Thanks in advance.
>
> AC
Not sure what result you were expecting.
Depending on where you expand your Taylor series, you might indeed get a
Puiseaux series result (that is, a series with fractional powers).
Below I use an example similar to yours above.
In[17]:= ee = 3*q^(1/3) + 7*q + 2*q^(1/2) + q^2 + 4*q^(7/3);
In[18]:= Series[ee, {q,1,3}]
2 3
61 (-1 + q) 239 (-1 + q) 649 (-1 + q) 4
Out[18]= 17 + ----------- + ------------- + ------------- + O[-1 + q]
3 36 648
In[19]:= Series[ee, {q,0,3}]
1/3 2 7/3 10/3 Out[19]= 3 q
+ 2 Sqrt[q] + 7 q + q + 4 q + O[q]
Mathematically we do not have a "function" y=f[q] in a neighborhood of
the origin, but rather a "multi-valued function" where distinct
branches of some equation g[y,q]==0 come together. One can show that
for well-behaved g (e.g. for g polynomial or analytic in each variable)
the set of points at which we cannot write y=f[q] is of measure zero
(and indeed contains no cluster point).
Daniel Lichtblau
Wolfram Research