How to plot the complex equation in 2D ...?

Questions
1

sans01.gif

Thank you for your cooperation.

2

ParametricPlot[{Re[Exp[I t]], Im[Exp[I t]]}, {t, 0, 2 Pi}]

3

Try these 3D plots:

Plot3D[Re[E^(x + I y)], {x, 0, 2 Pi}, {y, 0, 2 Pi}, PlotRange -> All]

Plot3D[Im[E^(x + I y)], {x, 0, 2 Pi}, {y, 0, 2 Pi}, PlotRange -> All]
4

How to plot this new curve ?

nou01.png
nou01.png216×349 3.22 KB

5

See attached notebook.

6

a beautiful presentation …

7

The trick to getting the vertical lines is to use Exclusions → None as a plot option:

zr[t_] := Re[Sqrt[E^(-7 I t)] + E^(I t)]
zi[t_] := Im[Sqrt[E^(-7 I t)] + E^(I t)]
ParametricPlot[{zr[t], zi[t]}, {t, -2, 5}, MaxRecursion -> 10,  Exclusions -> None]

8533parametricplot.jpeg
8533parametricplot.jpeg257×332 11.2 KB

8

Folks take a note of the new M10.1 function ReIm. With it code becomes simpler:

ParametricPlot[ReIm[Sqrt[E^(-7 I t)] + E^(I t)], {t, -2, 5}, Exclusions -> None]

Simon Cadrin, I suggest reading about proper posting of code on this website. It is easier for those who help you to copy code from your post. Equations in images are strongly discouraged.