       To verify Cauchy-Riemann relations in complex variable graphically

• To: mathgroup at smc.vnet.net
• Subject: [mg39176] To verify Cauchy-Riemann relations in complex variable graphically
• From: "Narasimham G.L." <google.news.invalid at web2news.net>
• Date: Sun, 2 Feb 2003 01:13:31 -0500 (EST)
• Reply-to: "Narasimham G.L." <mathman-3ospp+am18 at hotmail.com>
• Sender: owner-wri-mathgroup at wolfram.com

```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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