       Re: A Series test

• To: mathgroup at smc.vnet.net
• Subject: [mg72770] Re: [mg72765] A Series test
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 19 Jan 2007 01:09:21 -0500 (EST)
• References: <200701181102.GAA01494@smc.vnet.net>

```carlos at colorado.edu wrote:
> Just curious. Could somebody pls run this script on the
> latest Mathematica user version (I think it's 5.2) under
> Windows or Unix  and report the results:
>
> rho=(x+a*I)/(x-a*I); R=Abs[rho];
> s=Series[R,{x,0,4}];
> Print[FullSimplify[s,a>=0&&x>=0]//InputForm];
>
> My 5.0 answer (Mac G5 under OS 10.4.8) is
>
> SeriesData[x, 0, {1, (-2*I)/a, (-2*(1 + Derivative[Abs][-1]))/a^2,
> (((2*I)/3)*(3 + 6*Derivative[Abs][-1] - 2*Derivative[Abs][-1]))/
>     a^3, (2*(3 + 9*Derivative[Abs][-1] - 6*Derivative[Abs][-1] +
> Derivative[Abs][-1]))/(3*a^4)}, 0, 5, 1]
>
> The correct answer is 1. (The result with Simplify is more
> complicated.)  Thanks.

I'll leave the 5.2 run to others. Here is what happens in the
development kernel of Mathematica.

rho = (x+a*I)/(x-a*I);
absrho = Abs[rho];

In:= InputForm[s0 = Series[absrho, {x,0,4}]]
Out//InputForm= Abs[(I*a + x)/((-I)*a + x)]

In:= InputForm[s0b = FullSimplify[s0,Assumptions->a>=0&&x>=0]]
Out//InputForm= 1

Other possibile routes include using

absrho2 = Sqrt[rho*Conjugate[rho]];

or calling ComplexExpand explicitly to get your absolute value. This
last will give 1 immediately. Also will do that in version 5.2, if you
use Abs rather than Sqrt[rho*Conjugate[rho]].

Daniel Lichtblau
Wolfram Research

```

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