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MathGroup Archive 2004

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Re: complex analysis problem in mathematica 3.0

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48636] Re: complex analysis problem in mathematica 3.0
  • From: Sotirios Bonanos <sbonano at inp.demokritos.gr>
  • Date: Wed, 9 Jun 2004 04:16:53 -0400 (EDT)
  • Organization: National Technical University of Athens, Greece
  • References: <c9ec75$g9q$1@smc.vnet.net> <c9haqn$1b8$1@smc.vnet.net> <c9k4dl$fim$1@smc.vnet.net>
  • Reply-to: sbonano at inp.demokritos.gr
  • Sender: owner-wri-mathgroup at wolfram.com

BranasMan wrote:

> On Tue, 1 Jun 2004 07:21:59 +0000 (UTC)
> klishko at mail.ru (Alex Klishko)
> wrote:
>
> >I think, it confuses because of a variable w may also be a comlex number.
> >If w is a real try ComplexExpand -fuction.
> >For example:
> >
> >H = 1 / (s^3 + 2s^2 + 2s + 1) /. s -> \jw
> >ComplexExpand[%]
>
> w IS real,but i can't get mathematica to see that, and when i do
> ComplexExpand[%] i get the following:
>
> (1/Abs[1 + 2Iw - 2w^2 - Iw^3]^2 -
>     (2Iw)/Abs[1 + 2Iw - 2w^2 - Iw^3]^2 -
>     (2w^2)/Abs[1 + 2Iw - 2w^2 - Iw^3]^2 +
>     (Iw^3\)/Abs[1 + 2Iw - 2w^2 - Iw^3]^2)
>
> does anybody manage to get this thing right?
> i'm a little lost...

I have found it very useful to define a function Bar[x_] to give the
complex-conjugate of any expression x, and use it to define my own Re, Im
and Abs functions. The definition of Bar is in terms of a set of rules, which
can be modified to accommodate the symbols/expressions one uses.

Copy and paste the text beginning with  (*************)  below in a new
Mathematica notebook to see 2 examples of what I mean.
Using either definition, the functions myRe[h[I w]], myIm[h[I w]], where
h[s_] = 1/(s^3 + 2s^2 + 2s + 1), give the required simplified expressions.

S. Bonanos
http://www.inp.demokritos.gr/~sbonano/


              (*************)

              Notebook[{
              Cell[BoxData[
                  \( (**\ Define\ complex\ conjugate\ operation\ \*"\""Bar
\*"\"\<\>
              "**)
                    \)], "Input",
                FontColor->RGBColor[0, 0, 1]],

              Cell[BoxData[
                  \(SetAttributes[Bar, {Listable}]; \nBar[Bar[x_]] = x; \n
                  Bar[Complex[u_, v_]] := Complex[u, \(-v\)]; \n
                  Bar[x_Plus | x_Times | x_Power | x_Rule | x_Equal] :=
Bar/@x; \)],
                "Input"],

              Cell[BoxData[
                  \(\( (**\ Case\ I : \ All\ symbols\ used\ are\ real\ **) \n

                  \(Bar[x_] := x; \)\)\)], "Input"],

              Cell[BoxData[
                  \(\n\)], "Input"],

              Cell[BoxData[
                  \( (*\(*\ Use\ Bar\ to\ define\ Re\), \ Im, \ Abs\ **) \)],
"Input",
                FontColor->RGBColor[0, 0, 1]],

              Cell[BoxData[
                  \(\(myRe[x_] := Simplify[\((x + Bar[x])\)/2]; \)\)],
"Input"],

              Cell[BoxData[
                  \(\(myIm[x_] := Simplify[\((x - Bar[x])\)/\((2  I)\)];
\)\)],
              "Input"],

              Cell[BoxData[
                  \(\(myAbs[x_] := PowerExpand[Sqrt[Simplify[myRe[x]^2 +
myIm[x]^2]]]; \)\)],
              "Input"],

              Cell[BoxData[
                  \(\n\)], "Input"],

              Cell[BoxData[
                  \( (**\ EXAMPLES\ **) \)], "Input",
                FontColor->RGBColor[0, 0, 1]],

              Cell[CellGroupData[{

              Cell[BoxData[
                  \(myRe[\((x + I\ y)\)^3]\)], "Input"],

              Cell[BoxData[
                  \(x\^3 - 3\ x\ y\^2\)], "Output"]
              }, Open  ]],

              Cell[CellGroupData[{

              Cell[BoxData[
                  \(myIm[\((x + I\ y)\) + a^2/\((x + I\ y)\)]\)], "Input"],

              Cell[BoxData[
                  \(\(y\ \((\(-a\^2\) + x\^2 + y\^2)\)\)\/\(x\^2 + y\^2\)\)],
"Output"]
              }, Open  ]],

              Cell[CellGroupData[{

              Cell[BoxData[
                  \(myAbs[a \((x - a + I\ y)\) + I\ b \((y - b + I\ x)\)]\)],
"Input"],

              Cell[BoxData[
                  \(\@\(\((a\^2 - a\ x + b\ x)\)\^2 + \((\(-b\^2\) + a\ y +
b\
              y)\)\^2\)\)],
                 "Output"]
              }, Open  ]],

              Cell[BoxData[
                  \(\(\n\n\)\)], "Input"],

              Cell[BoxData[
                  \( (**\ Case\ II : \ Complex\ symbols\ have\ the\ form\
z[i_]\ **)
              \)],
                "Input",
                FontColor->RGBColor[0, 0, 1]],

              Cell[BoxData[
                  \(\(Clear[Bar]; \)\)], "Input"],

              Cell[BoxData[
                  \(SetAttributes[Bar, {Listable}]; \nBar[Bar[x_]] = x; \n
                  Bar[Complex[u_, v_]] := Complex[u, \(-v\)]; \n
                  Bar[x_Plus | x_Times | x_Power | x_Rule | x_Equal] :=
Bar/@x; \)],
                "Input"],

              Cell[BoxData[
                  \(\(Bar[x_] := x /; Head[x] =!= z; \)\)], "Input"],

              Cell[BoxData[
                  \(\n\)], "Input"],

              Cell[BoxData[
                  \( (**\ THEN\ **) \)], "Input",
                FontColor->RGBColor[0, 0, 1]],

              Cell[CellGroupData[{

              Cell[BoxData[
                  \(myRe[z[1] + I\ z[2]]\)], "Input"],

              Cell[BoxData[
                  \(1\/2\ \((Bar[z[1]] - I\ Bar[z[2]] + z[1] + I\ z[2])\)\)],
"Output"]
              }, Open  ]],

              Cell[BoxData[
                  \(\(zTOxy = {z[1] -> x1 + I\ y1, z[2] -> x2 + I\ y2};
\)\)], "Input"],

              Cell[CellGroupData[{

              Cell[BoxData[
                  \(Together[%% /. %]\)], "Input"],

              Cell[BoxData[
                  \(x1 - y2\)], "Output"]
              }, Open  ]],

              Cell[CellGroupData[{

              Cell[BoxData[
                  \(myIm[z[1]^2 + I\ z[2]^2 /. zTOxy]\)], "Input"],

              Cell[BoxData[
                  \(x2\^2 + 2\ x1\ y1 - y2\^2\)], "Output"]
              }, Open  ]]
              },
              FrontEndVersion->"Macintosh 3.0",
              ScreenRectangle->{{0, 1024}, {0, 748}},
              WindowSize->{956, 721},
              WindowMargins->{{4, Automatic}, {Automatic, 2}},
              MacintoshSystemPageSetup->"\<\

00000000000000000NHh00000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000
              00000000000000000000000000000000\>"
              ]




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