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Re: symbolic division of series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113168] Re: symbolic division of series
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 16 Oct 2010 13:11:40 -0400 (EDT)
  • References: <201010151752.NAA23522@smc.vnet.net> <796AF5A7-165C-4C2D-B871-A9FCC9DEAE3E@mimuw.edu.pl>

Of course I should have started from the zeroth coefficients but the =
idea remains unchanged.

f=Sum[a[i] x^i,{i,0,10}]+O[x]^11;
g=Sum[b[i] x^i,{i,0,10}]+O[x]^11;

SeriesCoefficient[f/g, 9]


Andrzej Kozlowski

On 16 Oct 2010, at 08:05, Andrzej Kozlowski wrote:

> Here is an example that answer your question. Suppose you represent f =
and g as follows:
>
> f = Sum[a[i] x^i, {i, 1, 10}] + O[x]^11;
> g = Sum[b[i] x^i, {i, 1, 10}] + O[x]^11;
>
> In other words, you know the first 10 coefficients in the Taylor =
expansion of f and g at 0. Then you can get:
>
> SeriesCoefficient[f/g,5]
> (6)/b(1)-(a(5) b(2))/b(1)^2+(a(4) =
(b(2)^2/b(1)^2-b(3)/b(1)))/b(1)+(a(3) (-(b(2)^3/b(1)^3)+(2 b(3) =
b(2))/b(1)^2-b(4)/b(1)))/b(1)+(a(2) (b(2)^4/b(1)^4-(3 b(3) =
b(2)^2)/b(1)^3+(2 b(4) b(2))/b(1)^2+b(3)^2/b(1)^2-b(5)/b(1)))/b(1)+(a(1) =
(-(b(2)^5/b(1)^5)+(4 b(3) b(2)^3)/b(1)^4-(3 b(4) b(2)^2)/b(1)^3-(3 =
b(3)^2 b(2))/b(1)^3+(2 b(5) b(2))/b(1)^2+(2 b(3) =
b(4))/b(1)^2-b(6)/b(1)))/b(1)
>
> Of course this will only work for the coefficients that can be =
determined form the given information. So SeriesCoefficient[f/g, 9] =
works fine but
>
> SeriesCoefficient[f/g, 10]
>
> Indeterminate
>
>
> Andrzej Kozlowski
>
>
>
>
> On 15 Oct 2010, at 19:52, Leslaw Bieniasz wrote:
>
>>
>>
>> Hi,
>>
>> Suppose that I have two series expansions (Taylor or asymptotic
>> expansions) for functions f(x) and g(x). This means I know the =
formulae
>> for the series coefficients. Is there any way to use MATHEMATICA
>> to obtain symbolically the formulae for the coefficients
>> of the analogous series expansion of the ratio f(x)/g(x) ?
>> I need a possibly large number of the coefficients of such an =
expansion,
>> expressed as functions of the coefficients for f(x) and g(x).
>>
>> Leslaw
>>
>>
>



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