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Re: Problem with the Derivative of a Arg-function

• To: mathgroup at smc.vnet.net
• Subject: [mg48637] Re: Problem with the Derivative of a Arg-function
• From: klishko at mail.ru (Alex Klishko)
• Date: Wed, 9 Jun 2004 04:16:54 -0400 (EDT)
• References: <c9k37o$fbo$1@smc.vnet.net> <c9pfq9$t79$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

ab_def at prontomail.com (Maxim) wrote:

> The short answer is that the derivative of Arg is undefined: Arg is
> not an analytical function. Just consider what happens when you move
> along different directions, for example, from 1 to 1+eps and from 1 to
> 1+I*eps, with real eps.

Yes, you are right.
I think that when Mathematica take numerical derivative, it's ought to
show error message in case of complex argument (without using
ComplexExpand) .

> In[1]:=
> ComplexExpand[(Arg[z + h] - Arg[z])/h, z,
>     TargetFunctions -> Abs] //
>   Limit[#, h -> 0] & // FullSimplify
> 
> Out[1]=
> -Im[z]/Abs[z]^2

It's right for real argument, when Re'=1 or dRe[z]=dz
It common case it should be (-Im[z]*Re'[z]+Re[z]*Im'[z])/Abs[z]^2

Best wishes
Alex Klishko