rectplot[fff_,inputx_,inputy_, opts___] := ParametricPlot[Through[{Re,Im}[fff[x+I*y]]], {x,inputx[[1]], inputx[[2]]}, {y,inputy[[1]],inputy[[2]]}, opts, PlotStyle->None, Mesh->True];
discplot[fff_,inputx_,inputy_, opts___] :=ParametricPlot[Through[{Re,Im}[fff[r Exp[I t]]]],{r,inputx[[1]], inputx[[2]]},{t,inputy[[1]],inputy[[2]]},opts,PlotStyle->None, Mesh->True];

domain = rectplot[Identity,{0,2},{0,2}(*, PlotStyle->Red*) ]

f[z_] := 3z+4I+2

rectplot[f,{0,2},{0,2}]

g[z_] := z^2

squaregraph = rectplot[g,{0,2},{0,2}]

Show[domain, squaregraph, PlotRange->All]

discplot[Identity,{0,1},{0,Pi/2}]

discplot[f,{0,1},{0,Pi/2}]

discplot[g,{0,1},{0,Pi/2}]

The functions described in Sections 1.6 and 1.7 are graphed below.

rectplot[Exp, {0, 0.1}, {0, Pi}, PlotRange->All]

discplot[Exp, {0,1}, {0,2Pi}]

g[t_] := Exp[I t];
rectplot[g,{0,3},{0,2Pi}, PlotRange->All]

rectplot[Log, {0,3}, {0,3}]

rectplot[Log, {2.9,3}, {0,3}, Mesh->False]

discplot[Log, {0,1}, {0,Pi}]

(1)^I

N[%]

Section 1.7 Begins Here

rectplot[Sin, {0, Pi/2}, {0, 5}, PlotRange->All]

rectplot[Identity, {-Pi/2, Pi/2}, {-2, 2}]

rectplot[Sin, {-Pi/2, Pi/2}, {-1, 1}, PlotRange->All]

discplot[Identity,{0,1},{0,Pi/2}, PlotRange->All]

discplot[Sin,{0,1},{0,Pi/2}, PlotRange->All]

rectplot[Cos, {0, Pi/2}, {0, 1}]

discplot[ArcSin,{0,1},{-Pi,Pi}]