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MathGroup Archive 2002

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Re: Simplifying the *Individual Coefficients* in Series Expansions?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg35240] Re: [mg35214] Simplifying the *Individual Coefficients* in Series Expansions?
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 3 Jul 2002 05:15:03 -0400 (EDT)
  • References: <200207020612.CAA22353@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

AES wrote:
> 
> I have a long expression  f   that involves integers times various
> powers of symbols  b  and  x, i.e.
> 
>     f = ratio of two lengthy polynomials in  b  and  x
> 
> If I series expand this in  x , viz.
> 
>     fS  = Series[f,  {x, 0, 2}] // Normal
> 
> I get an answer in the form
> 
>     fS =  c1 x + c2 x^2
> 
> where the coefficients  c1  and  c2   in the resulting series expansion
> come out as rather messy expressions (ratios of polynomials).  In my
> problem, however, these coefficients actually happen to simplify
> substantially (since there are common factors in their numerators and
> denominators), and I'd like to have them in simplified form.  But if I
> write
> 
>     fS // Simplify
> 
> I'm back in lengthy polynomial form; and if I try something like
> 
>     fS = (Coefficient[fS, x] // Simplify) x +
>                                 (Coefficient[fS, x^2] // Simplify) x^2
> 
> I get an expression that looks great, but will not evaluate numerically.
> 
> Any easy way around this?


I do not understand what you mean by "will not evaluate numerically".
The expression that results looks like it should be mathematically
equivalent to fS.

You can perhaps do what you want using Collect on powers of x, with a
third argument specified to simplify the individual collected terms.
Below is an example.

In[25]:= rndpoly[deg_,var_] := Sum[Random[Integer,{1,10}]*var^j,
{j,0,deg}]

In[26]:= fnum = Sum[Random[Integer,{1,10}]*x^(j-1)*rndpoly[3,b], {j,8}];

In[27]:= fden = Sum[Random[Integer,{1,10}]*x^(j-1)*rndpoly[3,b], {j,8}];

In[28]:= fS = Normal[Series[fnum/fden, {x,0,2}]];

In[29]:= InputForm[Collect[fS, x, Simplify]]
Out[29]//InputForm= 
(5 + b + 10*b^2 + b^3)/(7 + 10*b + 9*b^2 + b^3) + 
 ((-26 + 180*b + 153*b^2 + 329*b^3 + 209*b^4 + 208*b^5 + 27*b^6)*x)/
  (6*(7 + 10*b + 9*b^2 + b^3)^2) - 
 ((-3672 - 8148*b - 10923*b^2 + 2927*b^3 + 16852*b^4 + 27818*b^5 + 
    20533*b^6 + 12421*b^7 + 2889*b^8 + 215*b^9)*x^2)/
  (12*(7 + 10*b + 9*b^2 + b^3)^3)


Daniel Lichtblau
Wolfram Research


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