       Re: RE: To verify Cauchy-Riemann relations in complex variable graphically

• To: mathgroup at smc.vnet.net
• Subject: [mg39228] Re: RE: To verify Cauchy-Riemann relations in complex variable graphically
• From: "Narasimham G.L." <google.news.invalid at web2news.net>
• Date: Tue, 4 Feb 2003 02:23:30 -0500 (EST)
• References: <b1l2k5\$1@smc.vnet.net>
• Reply-to: "Narasimham G.L." <manoos2p+8ammthma18 at hotmail.com>
• Sender: owner-wri-mathgroup at wolfram.com

```David Park:   Thanks, I get error msg by run
Show::gtype: CartesianMap is not a type of graphics.
Is it version realated? mine is V 2.2 -- Narasimham

> My friend Rip Pelletier pointed me to a better method to
> illustrate the
> Cauchy-Riemann relations for analytic functions. He
> pointed me to Tristan
> Needham's book 'Visual Complex Analysis'. In Chapter 5, Section I -
> Cauchy-Riemann Revealed, the CR conditions are related to the complex
> mapping.
>
> If a small patch of squares are mapped by an analytic
> function, then they go
> into another small patch in which all the squares have
> been amplified and
> rotated in exactly the same way.
>
> Fortunately, we have the ComplexMap package in Mathematica
> and can easily
> illustrate this for your functions. For example, for the
> Sin function...
>
> Needs['Graphics`ComplexMap`']
>
> With[
>     {x = 2,
>      y = 2,
>      del = 0.01,
>      f = Sin},
>     Show[GraphicsArray[{CartesianMap[
>             Identity, {x - del, x + del, del/5}, {y - del,
> y + del, del/5},
>             Axes -> False,
>             DisplayFunction -> Identity],
>
>           CartesianMap[
>             f, {x - del, x + del, del/5}, {y - del, y + del, del/5},
>             Axes -> False,
>             DisplayFunction -> Identity]}],
>       ImageSize -> 500]];
>
> A square patch maps into a rotated square patch. Just
> change f and/or the
> mapping points for other cases. Use a pure function for z^2.
>
> For a case that is not analytic, and so the CR relations
> do not hold, use
> f = # + 2Abs[#] &. The squares go to parallelograms.
>
> David Park
>
> From: Narasimham G.L. [mailto:google.news.invalid at web2news.net]
To: mathgroup at smc.vnet.net
>
> Is it possible to have a semi transparent view of surfaces
> so that one
> may verify slopes by ParametricPlot3D for Cauchy-Riemann relations?
> The following is program for 3 functions Z^2, Z^3, Sin[Z].It was
> expected to check slopes at the line of intersection of Re
> and Im parts.
>
> R1=x^2-y^2 ; I1= 2 x y ;
> z2r=Plot3D[R1 , {x,-Pi,Pi},{y,-Pi,Pi} ];
> z2i=Plot3D[I1 , {x,-Pi,Pi},{y,-Pi,Pi} ];
> Show[z2r,z2i] ; 'Top view >> Re,Im Intxn';
> Plot[{x ArcTan[-Sqrt+1],x ArcTan[Sqrt+1]}, {x,-Pi,Pi} ];
>
> R3=x^3 - 3 x y^2 ; I3= 3 x^2 y - y ^3 ;
> z3r=Plot3D[R3 , {x,-Pi,Pi},{y,-Pi,Pi} ];
> z3i=Plot3D[I3 , {x,-Pi,Pi},{y,-Pi,Pi} ];
> Show[z3r,z3i] ; 'Top view >> Re,Im Intxn';
> Plot[{x,x (-Sqrt+2) , x (-Sqrt-2) }, {x,-Pi,Pi} ];
>
> R2=Cosh[y] Sin[x] ; I2=Sinh[y] Cos[x] ;
> scr=Plot3D[R2,{x,-Pi/2,Pi/2},{y,-Pi/2,Pi/2}];
> sci=Plot3D[I2,{x,-Pi/2,Pi/2},{y,-Pi/2,Pi/2}];
> Show[scr,sci]; 'Top view >>  Re,Im Intxn';
> Plot[{ArcTanh[Tan[x]]},{x,-Pi/2,Pi/2 }];
--
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