       Re: Re: Differentiate of Abs[x] ???

• To: mathgroup at smc.vnet.net
• Subject: [mg53120] Re: [mg53087] Re: Differentiate of Abs[x] ???
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Fri, 24 Dec 2004 05:59:07 -0500 (EST)
• References: <cq0sv8\$2h8\$1@smc.vnet.net> <200412231258.HAA21085@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On 23 Dec 2004, at 21:58, Peter Pein wrote:

> idyllic wrote:
>> Dear all,
>>   If I have a complex-valued funciton Z=Z(x1,x2,y) where x1,x2,y are
>> all real varables. Now I demand that |Z|^2 are local minimums at y=y1
>> and y=y2 with some choices of x1 and x2. (where |...| denotes absolute
>> value.)
>>
>>   My idea is that, I differentiate |Z|^2 w.r.t y, then subtitute y1
>> and y2 into this equation and demand each equation 0; finally solve
>> the two simultaneous equations.
>>
>>   But I found a big problem that Mathematica can't approach the
>> differentiation of absolute values? Another question, is my
>> method right? and is the differentiation of absolute value very hard
>> to approach mathematically?
>> Thanks for every ideas.           Sincerely
>>
> Try TagSet :
>
> Unprotect[Derivative];
> Derivative /: Derivative[Abs] := Sign;
> Protect[Derivative];
>
> In:=
> D[Abs[Z[x1, x2, y]]^2, y]
> Out=
> 2*Abs[Z[x1, x2, y]]*Sign[Z[x1, x2, y]]*Derivative[0, 0, 1][Z][x1, x2,
> y]
>
> Peter
>
> --
> Peter Pein
> Berlin
>
>
>
This is  wrong  since Abs[] is not differentiable in the complex plane.
The definition of the derivative of Abs is valid only for real
arguments, but in that case it is in any case better to use Sqrt[x^2]