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 >> >> >

**References**:**symbolic division of series***From:*Leslaw Bieniasz <nbbienia@cyf-kr.edu.pl>