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 1.1

DisplayTogether[
Plot[
          Cos[Log[x]/x]/x, {x, 0.01,  .1},  
          Frame -> True,  
FrameTicks -> {Range[0, 
                1,   .2], Range[-100, 100, 50],  None,  
              None}, 
Axes -> False,  
          AspectRatio  ->   .5,  PlotStyle ->  Thickness[th], 
          TextStyle -> {FontSize -> fs,  
              FontFamily -> Times},  
          PlotRange -> {{- .001, 1}, {-100.1, 100}},  
          PlotLabel -> ("Integrand of Problem 1 with ϵ\
 = 0.01"; None),  PlotPoints -> 1000,  
          MaxBend -> 1], 
Plot[
          Cos[Log[x]/x]/x, {x, 0.1, 1},  MaxBend -> 1,  
          PlotStyle ->  Thickness[th]],  
        ImageSize -> width];

Figure 1.2 and Book Cover

fix10[s_String]  :=  
        DisplayForm[
          StyleBox[s,  FontFamily  ->  "Times", 
            FontSize -> fsbig]];
f[x_, y_] := (z = x + I y; 
          Re[(z^((I/z) - 1))]);
h[
          r_, θ_] := (z = 
            r (E^(I θ)); 
          Re[(z^((I/z) - 1))]);

    
Off[ParametricPlot3D::"pplr"];
LiftToViewPoint[pt_, vp_, br_, pr_, 
        lf_] := (vec = (Apply[Plus, pr, {1}]/
                2 + (-vp*Apply[Subtract, pr, {1}]) Max[br]/br - 
              pt); 
pt + 
          lf*vec/Sqrt[Norm[vec]^2])

pr = {{ .0, 1.001}, {- .02,  .27}, {-6.5;  -40, 6.8; 12; 
          40}}; vp = {- .1;  - .05, -2.4, 0.1}; 
    br = {1, 3,  .4};
\(\(cc1 = 
        Block[{$DisplayFunction\  = \ Identity}, 
          ParametricPlot3D[
{r^3\ Cos[θ^2], \ 
              r^3\ Sin[θ^2], 
              If[θ ≤ 20\ \ Degree\  && \ 
                  r\  < \ \(( .02)\)^\((1/3)\), ∞, 
                h[r^3, θ^2]]}, \ {r, \(( .001)\)^\((1/3)\), \ 
              1\ \@2}, \ {θ, \ 0, \ π/2}, 
            PlotPoints -> {200; 300, 40}, \ Boxed -> False, 
            AxesLabel\  -> {"x", \ "y", \ \*""|\!\(z\^\(I/z\\  - \\ 1\)\)|""}, \ 
            TextStyle -> {FontFamily\  -> "Times", \ 
                FontSize -> 
                  fs}, 
ViewPoint\  -> \ vp, \ 
            PlotRange -> \((pr)\), ImageSize -> width\ 1/2, \ 
            AxesEdge\  -> \ {{1, \(-1\)}, \ {1, \(-1\)}, \ {1, \(-1\)}}, 
            BoxRatios -> br]];\)\)

pp = 
          Cases[Plot[(1/x) Cos[Log[x]/x], {x, 0.069;  .025, 1 -  .01}, 
              DisplayFunction -> Identity, MaxBend -> 2, 
              PlotPoints -> 450], _Line, ∞];

pr = {{ .0, 
          1.001}, {- .02; - .03,  .27;  .29}, {-40, 40}}; 
    vp = { - .05, -2.4, 0.13}; 
    br = {1, 3,  .4};
fix[p_, amt_] :=  
        LiftToViewPoint[p, vp, br, pr,  .01;  .05; 
          amt];
im=Show[Graphics3D[{EdgeForm[],  
                Cases[cc1,  _Polygon,  ∞], 

{Hue[0], 
                  Thickness[ .006;   .0055], 
pp  /.   {x_,  y_}  :>  
                      fix[{x, 0, 
                          y},  .05], 
Thickness[ .0052], 
                  RGBColor[1, 1, 0], 
                  Line[Table[
                      fix[{xx, yy = xx (1 - xx), f[xx, yy]},  .8], {xx, 
                        0.001, 
                        1, (1 - 0.001)/
                          100}]], 
{GrayLevel[0], 
                    PointSize[ .012;   .011], 
                    Point /@  {{1, 0, 1}, {0, 0, 0}}}}}],
 BoxRatios  ->  br,Boxed -> False,
ViewPoint  ->  vp, PlotRange -> pr,Axes -> True,
\(Ticks -> {{1,  .8,  .2,  .4,  .6}, {{0, ""}, { .15, ""}}, Range[\(-30\), 30, 15]}\),
\(TextStyle -> {FontFamily\  -> \ "Times", \ FontSize -> fs}\),
Epilog -> {Text["Re(z)",{ .875,  .26} ,
      TextStyle ->  { 
                          FontFamily  ->  "Times",  
                          FontSize -> fsbig}],
Text["Im(z)",{ .96,  .34} ,TextStyle ->  { 
                          FontFamily  ->  "Times",  
                          FontSize -> fsbig}],
Text["0", { .975,  .28} , 
                    TextStyle ->  { FontFamily  ->  "Times",  
                        FontSize -> fs}],
Text["0.15", { .89,  .38} , 
                    TextStyle ->  { FontFamily  ->  "Times",  
                        FontSize -> fs}]},
AxesLabel  -> (None), AxesEdge  ->  {{-1, -1},  {1, -1},  {-1, -1}} ,ImageSize ->  1.3 width];

Version for Cover

Show[im,  Axes -> False,  
        AxesLabel -> {"", "", ""},  Ticks -> None,  
        Epilog -> {},  ImageSize -> 2.3 width];

Figure 1.3

Clear[z]; z[t_] := t + I t (1 - t);
dz[t_] := 1 + I - 2 I t;
f[t_] := 
      Re[z[t]^((I/z)[t] - 1) dz[t]]
Plot[f[t],  {t, 0, 1},  
        PlotRange -> { {- .0001,  1},  {-0.2001,  1}},  
        Frame -> True,  Axes -> False,  
        PlotStyle -> Thickness[th], 
        FrameTicks ->  {Range[0, 1,   .2],  
            Range[- .2,  1,   .2],  {}, {}},  
        TextStyle -> {FontSize -> fs, FontFamily -> "Times"},  
        MaxBend -> 1,  PlotPoints -> 100,  
        ImageSize -> width];

Figure 1.4

Plot[Cos[t E^t],  {t, 0, 4},  Frame -> True,  
        Axes -> False,  
 
        PlotRange -> { {- .0001, 4},  {-1.2,  
              1.2}}, 
 PlotStyle -> Thickness[th], 
        FrameTicks ->  {Range[0, 4,   .5],  
            N@Range[-1,  1,   .5],  {}, {}},  
        TextStyle -> {FontSize -> fs, FontFamily -> "Times"},  
        MaxBend -> 1,  PlotPoints -> 1000,  
        PlotStyle -> Thickness[ .0004],  
        ImageSize -> width];

Figure 1.5

\(\(Plot[{6, 18, 25, 
              15} . \(1\/\((1 + t)\)\^Range[3, 6]\)\ E\^\(\(-3\)\ t\)\ Sin[t\ E\^t], {t, 0, 12}, 
        Frame -> True, Axes -> False, 
\ 
        PlotStyle -> Thickness[th], 
        PlotRange -> {{\(- .0001\), 12}, {\(- .5\), 4. }}, 
        FrameTicks -> {Range[0, 12, 2], N[Range[0, 4, 1]], {}, {}}, 
        TextStyle -> {FontSize -> fs, FontFamily -> "Times"}, 
        MaxBend -> 1, PlotPoints -> 100, 
        PlotStyle -> Thickness[ .0004], \ \ ImageSize -> 
          width];\)\)

Figure 1.6

\(\(Plot[
        Evaluate[{\(-1\), 1} ArcCosh[\(-2\)\ \((\(-1\) + x)\)]], \ {x, \ 
          1 - Cosh[\(-2.8125\)]/2, 0.5}, \ 
        PlotRange -> {{\(-3.500\), \  .6}, \ {\(-3.001\), 3}}, \ 
        Frame -> True, \ Axes -> False, 
        PlotStyle -> 
          Thickness[
            th], 
FrameTicks -> {Range[\(-3.5\), \  .5,  .5], \ Range[\(-3\), 3], \ None, \ None}, 
        TextStyle -> {FontSize -> fs, FontFamily -> "Times"}, 
        Epilog\  -> \ {PointSize[ .013], 
pts = 
              Table[Point[{1 - 1/2  Cosh[tt], tt}], \ {tt, \(-2.8125\), 
                  2.8125, 1/16}]}, \ \ ImageSize -> width];\)\)

Figure 1.7

f[u_]  :=  
      Evaluate[D[ProductLog[u], u] Cos[u]]

    
\(\(Plot[f[u], \ {u, 0, 20}, Frame -> True, Axes -> False, 
        PlotRange -> {{\(- .0001\), 20}, {\(- .200\), 1. }}, 
        FrameTicks -> {{{π\/2, "π/2"}, {\(3  π\)\/2, "3π/2"}, {\(5  π\)\/2, "5π/2"}, {\(7  π\)\/2, "7π/2"}, {\(9  π\)\/2, "9π/2"}, {\(11  π\)\/2, "11π/2"}}, N[Range[\(- .2\), 1,  .2]], {}, {}}, 
        TextStyle -> {FontSize -> fs, FontFamily -> "Times"}, 
        MaxBend -> 1, PlotPoints -> 100, 
        AspectRatio ->  .55, 
GridLines -> {{}, {{0, {Thickness[th], Dashing[{ .01,  .02}]}}}}, 
        PlotStyle -> Thickness[th], \ 
        ImageSize -> width];\)\)