Re: Re: Problem with the Derivative of a Arg-function

• To: mathgroup at smc.vnet.net
• Subject: [mg48556] Re: [mg48514] Re: Problem with the Derivative of a Arg-function
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 5 Jun 2004 07:18:56 -0400 (EDT)
• References: <200406040849.EAA23630@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On 4 Jun 2004, at 17:49, Alex Klishko wrote:

>> On 2 Jun 2004, at 17:21, Andrzej Kozlowski wrote:
>>
>> I am not sure what you mean by "fase"? Is that just Arg[E^(-I*x)]? Are
>> you saying that this is -Re[x] for all complex x? Well, that certianly
>> is not true, even for real ones!
>
> You are right, Arg[E^(-I*x)] is equal to -x +2*Pi*n, where n is an
> integer number so, that -Pi<%<Pi.

I suppose you mean -Re[x]+2*Pi*n. But of course for complex  x  Re[x]
is not differentiable (as a function of the complex variable x !) so
Mathematica is quite right not to return -1.
Of course

D[ComplexExpand[Arg[E^((-I)*x)], {x}],Re[x]]//Simplify

-1

but that is a completely differnt thing!

Andrzej Kozlowski

>
> But I need an Arg's derivative.
> As 2*Pi*n is the constant so the derivative doesn't depend on it.
> So the derivative must be -1.
>
> By the way,
>  f[x_] = ComplexExpand[Arg[E^(-I*x)]];  N[D[f[x], x]]  /.  x -> x0
>
> gives -1, for any x0  (if you substitute instead x0 any real number)
>

```

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