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help me to clarify the transformation of the functions by using Mathematica enter image description here

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1 Answer 1

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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}]]

enter image description here

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

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

enter image description here

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}]]

enter image description here

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