Re: Expansion Coefficients in Multivariate Series Expansions?

*To*: mathgroup at smc.vnet.net*Subject*: [mg26599] Re: [mg26579] Expansion Coefficients in Multivariate Series Expansions?*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Thu, 11 Jan 2001 10:39:11 -0500 (EST)*References*: <200101090651.BAA00200@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

"A. E. Siegman" wrote: > > Given an expression that is the sum of products of integer powers of > three variables (say x,y,z) multiplied by numerical coefficients: > > f = c000 + c100 x + c010 y + c001 z + c110 x y + c211 x^2 y z + . . . > > where the cijk's are all real numbers, and f may or may not be in > Normal (or other) form, how do I find the coefficient of a particular > term, e.g. c100 for the x term (*not* c1000 + c110 y), or ckmn for the > x^k y^m z^n term? > > Neither Coefficient nor SeriesCoefficient seems to do this -- and the > SeriesCoefficient[expr, {n1,n2,n3,..}] syntax seems to be mentioned but > neither explained or illustrated in the on-line Help. > > Thanks, siegman at stanford.edu For nonzero powers Coefficient will work just fine. To use SeriesCoefficient you first need a series. In[59]:= f = c000 + c100 x + c010 y + c001 z + c110 x y + c211 x^2 y z ; In[60]:= ee = Series[f, {x,0,3}, {y,0,3}, {z,0,3}]; In[62]:= SeriesCoefficient[ee, {0,0,0}] Out[62]= c000 Another possibility is to iterate Coefficient. In[63]:= Coefficient[Coefficient[Coefficient[f, x, 0], y, 0], z, 0] Out[63]= c000 Or you could write your own code. Below is a simple procedural way, taking derivatives and setting variables to zero. In[64]:= myCoefficient[ee_, vars__] := Module[ {len, vlist={vars}, res=ee}, Do[ res = D[res, vlist[[j]]] /. vlist[[j,1]]->0, {j,Length[vlist]}]; res ] In[66]:= myCoefficient[f, {x,0}, {y,0}, {z,0}] Out[66]= c000 A functional programming approach might use Fold. In[72]:= myCoefficient2[ee_, vars__] := Fold[(D[#1,#2] /. #2[[1]]->0)&, f, {vars}] In[73]:= myCoefficient2[f, {x,0}, {y,0}, {z,0}] Out[73]= c000 Daniel Lichtblau Wolfram Research

**References**:**Expansion Coefficients in Multivariate Series Expansions?***From:*"A. E. Siegman" <siegman@stanford.edu>