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Re: Complex diagram

• To: mathgroup at smc.vnet.net
• Subject: [mg123465] Re: Complex diagram
• From: Chris Young <cy56 at comcast.net>
• Date: Thu, 8 Dec 2011 05:24:32 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jbni9o$48v$1@smc.vnet.net>

On 2011-12-07 11:22:32 +0000, é?? å?? said:

> Can we draw complex funtions's diagram in Mathematica?
> 
> For example, draw the picture of z=2*x+I*3*y

This isn't an analytic function, which boils down to the fact that it's 
not a mapping that preserves angles. So the two sets of contour lines 
below won't be perpendicular. Note that if we just use z as the 
variable, we'll be guaranteed an analytic function.

http://mathworld.wolfram.com/AnalyticFunction.html
Calculus and Analysis > Complex Analysis > General Complex Analysis >

Analytic Function
	
A complex function is said to be analytic on a region  if it is complex 
differentiable at every point in . The terms holomorphic function, 
differentiable function, and complex differentiable function are 
sometimes used interchangeably with "analytic function" (Krantz 1999, 
p. 16). Many mathematicians prefer the term "holomorphic function" (or 
"holomorphic map") to "analytic function" (Krantz 1999, p. 16), while 
"analytic" appears to be in widespread use among physicists, engineers, 
and in some older texts (e.g., Morse and Feshbach 1953, pp. 356-374; 
Knopp 1996, pp. 83-111; Whittaker and Watson 1990, p. 83).

If a complex function is analytic on a region , it is infinitely 
differentiable in . A complex function may fail to be analytic at one 
or more points through the presence of singularities, or along lines or 
line segments through the presence of branch cuts.

A complex function that is analytic at all finite points of the complex 
plane is said to be entire. A single-valued function that is analytic 
in all but possibly a discrete subset of its domain, and at those 
singularities goes to infinity like a polynomial (i.e., these 
exceptional points must be poles and not essential singularities), is 
called a meromorphic function.


f[x_, y_] := 2 x + 3 y I

g[z_] := ((2 + 3)/2) z

Column[{
  Row[{
    ContourPlot[
     Arg[f[x, y]],
     {x, -2, 2}, {y, -2, 2},
     ColorFunction -> (f \[Function] Hue[f, 0.5, 1]),
     Contours -> 11],

    ContourPlot[
     Abs[f[x, y]],
     {x, -2, 2}, {y, -2, 2},
     Contours -> 10
     ]
    }],

  Row[{
    ContourPlot[
     Arg[g[x + y I]],
     {x, -2, 2}, {y, -2, 2},
     ColorFunction -> (f \[Function] Hue[f, 0.5, 1]),
     Contours -> 11],

    ContourPlot[
     Abs[g[x + y I]],
     {x, -2, 2}, {y, -2, 2},
     Contours -> 10
     ]
    }]
  }]


For a 3D plot of the modulus:

ContourPlot3D[
 z - Abs[f[x, y]] == 0,
 {x, -2, 2}, {y, -2, 2}, {z, 0, 4},
 MeshFunctions -> {({x, y, z} \[Function]
     z), ( {x, y, z, f} \[Function] f)}
 ]