Re: symbolic division of series

*To*: mathgroup at smc.vnet.net*Subject*: [mg113169] Re: symbolic division of series*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 16 Oct 2010 13:11:51 -0400 (EDT)*References*: <201010151752.NAA23522@smc.vnet.net>

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>