help me to clarify the transformation of the functions by using Mathematica

Question

Hello! Previous related questions include plot of $z \mapsto z^2$, plot of $z \mapsto 1/z$, plot of $z \mapsto \log(z)$, plot of the Cayley map and plot of some other map. Many approaches use ParametricPlot. Have you tried to adapt one of those approaches to your map? — user293787, Jul 17 '22 at 08:13
Incidentally, does plot of $z \mapsto 1/z$ answer your question, or are you looking for something else? — user293787, Jul 17 '22 at 08:27

score 1 · Answer 1 · answered Jul 17 '22 at 20:01

We first define two functions: linesx and linesy that draw the x and y lines in the given strip:

xmax = 1;
ymin = 1/4; ymax = 1/2;
linesx[x_] = Table[{x, y}, {y, 1/4, 1/2, 1/16}];
linesy[y_] = Table[{x, y}, {x, -xmax, xmax, 0.1}];
Show[ParametricPlot[linesx[x], {x, -xmax, xmax}], 
 ParametricPlot[linesy[y], {y, ymin, ymax}]]

To get the picture of these lines under the mapping 1/z we use "ComplexExpand":

ReIm[1/(x + I y)] // ComplexExpand

We use this to define the functions describing the pictures of the x/y lines:

zlinesx[x_] = 
  Table[{x/(x^2 + y^2), -(y/(x^2 + y^2))}, {y, 1/4, 1/2, 1/16}];
zlinesy[y_] = 
  Table[{x/(x^2 + y^2), -(y/(x^2 + y^2))}, {x, -xmax, xmax, 0.1}];
Show[ParametricPlot[zlinesx[x], {x, -xmax, xmax}], 
 ParametricPlot[zlinesy[y], {y, ymin, ymax}]]

help me to clarify the transformation of the functions by using Mathematica

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