Graphics Settings

<< Miscellaneous`Units`;
Remove[Gray];
<< Graphics`Master`;
Off[General::spell1];
fs = 7;
fsbig = 9;
width  =  Convert[8.64 Centi Meter,  Point]/Point;
th  =  0.0008;

Figure 3.1

Show[Block[{$DisplayFunction  =  
                Identity}, 
Plot[{Sqrt[-1/4 + 
                    x^2], -Sqrt[-1/4 + x^2]},  {x, 1/2, 
                4}, 
MaxBend -> 1]],

Graphics[  
{PointSize[ .02], Point  /@  {{ .5,  
                      1.5}, 
{ .5,  -1.5},  {1.5,  
                      2.15}, {1.5,  -2.15}},
Point[{xx = ((Sqrt[(51/2)])/2), 
                    yy =  Sqrt[-1/4 + xx^2]}],
P = {xx, yy}; 
                Point[P {1, -1}],

Circle[{0, 0},  
                  3.5,  {-ArcTan[yy/xx], ArcTan[yy/xx]}],
Text[ℂ,  {-1.4, 2.4},  TextStyle  ->  FontSize -> 20],
Text["poles of f", { .76, 1.86}, TextStyle  -> {FontFamily  ->  
                          Times,  FontSize -> fsbig}],
{{GrayLevel[1], 
                        Disk[{#, 0},   .05]}, 
                      Circle[{#, 0},   .05]} &  /@  Range[-1, 5],
Text["r", {3.7, 0.1}, \(TextStyle -> {FontFamily\  -> \ "Times", \ FontSize -> fsbig}\)],\(Text[\*""\!\(𝒞\_1\)"", \ {2.1, 
                    1.7}, \ 
                  TextStyle -> {FontFamily\  -> \ "Times", \ 
                      FontSize -> fsbig}]\),
Text["𝒞", {3.15, 2.8}, TextStyle -> {FontFamily  ->  "Times\
",  FontSize -> fsbig}],
Text["(Subscript[(𝒞), 2]): circular arc\n  of radius r->∞", {3.1, 1.9}, {-1, 0},TextStyle -> {FontFamily  ->  "Times\
",  FontSize -> fsbig}],
Text["P", P + {- .15,  .15}, TextStyle -> {FontFamily  ->  "Times\
",  FontSize -> fsbig}],Text["\!\(\(\*
StyleBox["P",
FontSlant->"Italic"]\)\&_\)", P {1, -1} + {- .23, - .2}, TextStyle -> {FontFamily  ->  "Times\
",  FontSize -> fsbig}],

Polygon[{ss1 = {xx = 3.35,  
                        Sqrt[3.5^2 - xx^2]};  
                    ss2 = {xx = 3.28,  
                        Sqrt[3.5^2 - xx^2]}; 
perp =  
                      Reverse[ss2 - ss1] {1, -1}; ss1 + perp  .3 , ss2, 
                    ss1 - perp  .3}],

Polygon[{ss1 = {xx = 1.75,  
                        Sqrt[- .5^2 + xx^2]};  
                    ss2 = {xx = 1.6,  
                        Sqrt[- .5^2 + xx^2]}; 
perp =  
                      Reverse[ss2 - ss1] {1, -1}; ss1 + perp  .3 , ss2, 
                    ss1 - perp  .3}],

Polygon[{ss1 = {xx = 
                          1.75,  -Sqrt[- .5^2 + xx^2]};  
                    ss2 = {xx = 
                          1.9,  -Sqrt[- .5^2 + 
                              xx^2]}; 
perp =  
                      Reverse[ss2 - ss1] {1, -1}; ss1 + perp  .3 , ss2, 
                    ss1 - perp  .3}]

}
],PlotRange -> {{-2, 5.6}, {-3, 3}},   AspectRatio -> Automatic, Axes -> True, Frame -> True, 
\(FrameTicks -> {Range[\(-1\), 
                5], \ 
\({#, \ 
                    StringForm["``I", \ #] /. \ {StringForm["``I", \ 
                            0]\  -> \ "0", 
StringForm["``I", \ 
                            1]\  -> \ "I", 
StringForm["``I", \ \(-1\)]\  -> \ "-I"}} &\)\  /@ \ Range[\(-2\), \ 3]\ , \ None, \ None}\), TextStyle -> {FontSize -> fs,  
              FontFamily  -> "Times"}, ImageSize -> width 1.05];