help me to clarify the transformation of the functions by using Mathematica 
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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}]] 
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}]] 
ParametricPlot. Have you tried to adapt one of those approaches to your map? $\endgroup$