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MathGroup Archive 2007

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Re: Real/Comlex function problem with D

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74242] Re: Real/Comlex function problem with D
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Thu, 15 Mar 2007 05:03:26 -0500 (EST)
  • References: <et8e29$pqb$1@smc.vnet.net>

On Mar 14, 11:08 am, b... at ANTYSPAM.ap.krakow.pl wrote:
> Hi,
> I have a complex function F[x]
> When i defined:
> ff[x_]:=Arg[F[x]]   (* i tried ComplexExpand[Arg[F[x]]] too *)
>
> I obtainned good Plot[ff[x]] - its real - (no complex)
>
> When i try
> ff1[x_]:=D[ff[x],x];
>  I found ff1 is Complex funtion again !!
>
> Why ?
>
> (i tried Evaluate; Element[x,Reals]; FullSimplify - always the same)
>
> Regards, Olaf

Is it a particular reason that the derivative of Arg[F[x]] should not
be complex?

Print["a complex function"]
F[x_] = x*Exp[3*I*x]
Print["the argument of the complex function"]
ff[x_] = Arg[F[x]]
Print["plots of the real and imaginary\npart of the complex function
\nalong with plot of the argument"]
Show[GraphicsArray[Block[{$DisplayFunction = Identity},
(Plot[#1[F[x]], {x, 0, 2*Pi}, PlotLabel -> #1[F[x]]] & ) /@ {Re,
Im}]],
   ImageSize -> 400];
Show[Block[{$DisplayFunction = Identity}, (Plot[ff[x], {x, #1[[1]],
#1[[2]]}] & ) /@ Partition[Range[0, 2*Pi, Pi/3], 2, 1]],
   Axes -> False, Frame -> {True, True, False, False}, ImageSize ->
200, PlotLabel -> ff[x]];
Print[StyleForm["note the finite discontinuities at n*Pi/3", FontColor
-> Red]]
Print["the derivative of Arg[F[x]] is complex function"]
ff1[x_] = D[ff[x], x]
Print["the real and the imaginary part"]
(ComplexExpand[#1[D[ff[x], x]]] & ) /@ {Re, Im}
Print["the plots of them\nred\[Implies]Re, blue\[Implies]Im"]
Plot[Evaluate[%%], {x, 0, 2*Pi}, PlotStyle -> {Red, Blue}];

Best Regards
Dimitris

P.S.
 The result from Arg[z] is always between -Pi and Pi.
 Arg[z] has a branch cut discontinuity in the complex z plane running
from -Infinity to 0.

Copy paste the following to a notebook

Cell[TextData[ButtonBox["Arg from www.functions.wolfram.com",
  ButtonData:>{
    URL[ "http://functions.wolfram.com/ComplexComponents/Arg/";],
None},
  ButtonStyle->"Hyperlink"]], "Text"]

select the cell and press Shift+Ctrl+E
Then with your mouse click the the hyperlink to go to a relevant web
page for more information.



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