Re: Bug in Series with NonCommutativeProduct?

*To*: mathgroup at smc.vnet.net*Subject*: [mg122198] Re: Bug in Series with NonCommutativeProduct?*From*: D J G C <dundjoh at googlemail.com>*Date*: Thu, 20 Oct 2011 07:43:50 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201110130748.DAA02390@smc.vnet.net> <j79160$i4b$1@smc.vnet.net>

On Oct 14, 11:59 am, Daniel Lichtblau <d... at wolfram.com> wrote: > On 10/13/2011 02:48 AM, D J G C wrote: > > >>Series[(A h) ** (B h), {h, 0, 2}] > > > ... + A B h^2+O[h]^3 > > > The problem is the appearance of a commutative product A B. It should > > instead be A ** B > > > However applying the D gives the correct answer: > > >> D[(A h ) ** (B h), {h, 2}] > > > 0**(B h)+2 A**B+(A h)**0 > > It's just doing a "blind" Taylor expansion. One can see this by > replacing NonCommutativeMultiply with some (undefined) function f: > > In[242]:=Series[f[(A h), (B h)], {h, 0, 2}] > > Out[242]= SeriesData[h, 0, { > f[0, 0], B Derivative[0, 1][f][0, 0] + A Derivative[1, 0][f][0, 0], > Rational[1, 2] ( > B^2 Derivative[0, 2][f][0, 0] + 2 A B Derivative[1, 1][f][ > 0, 0] + A^2 Derivative[2, 0][f][0, 0])}, 0, 3, 1] > > Daniel Lichtblau > Wolfram Research A series always is a "blind" Taylor expansion. The bug is, that in this "blind" expansion it gets the second derivative wrong, by ignoring the possibility for a non-commutative product. The D[] is aware of this. Why not use its result in the expansion (which by the way is the definition of a series)? This is a bug, as Mathematica is a mathematical software, offering the (a) non-commutative product, which part of it does not handle. At least the user has to be warned.

**References**:**Bug in Series with NonCommutativeProduct?***From:*D J G C <dundjoh@googlemail.com>