       Re: Bug in Series with NonCommutativeProduct?

• To: mathgroup at smc.vnet.net
• Subject: [mg122111] Re: Bug in Series with NonCommutativeProduct?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 14 Oct 2011 05:54:40 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201110130748.DAA02390@smc.vnet.net>

```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:= Series[f[(A h), (B h)], {h, 0, 2}]

Out= 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

```

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