A geometric interpretation of a function of a complex variable with Mathematica?

Question

How to write code in math to see the result as follows:

As you can see, we have a complex function w=f(z) , where z=x+iy and w=u+iv .

In this example, w=sin z , and we see that the image of the vertical line x=c is a hyperbola. (Left)

and the image of the horizontal line y=c is elliptical. (On right)

This picture is taken from the book “Complex Variables and Applications” by James Ward Brown, Ruel Vance Churchill, 8th edition: pages 331 and 333 or pages of the third edition 96-97

Answer 1

Something like that?

 ClearAll[u, v, f]; f[z_] := Sin[z] u[x_, y_] := Re@f [x + I*y]; v[x_, y_] := Im@f [x + I*y]; 

EDIT: It just produces everything. If you just want to see what happens for one path parallel to an imaginary axis, try

 ParametricPlot[{u[5, y], v[5, y]}, {y, -3, 3}] 

or for the same parallel with the real try axis

 ParametricPlot[{u[x, 1], v[x, 1]}, {x, -3, 3}] 

EDIT2: Interactive:

 ClearAll[u, v, f]; f[z_] := Sin[z] u[x_, y_] := Re@f [x + I*y]; v[x_, y_] := Im@f [x + I*y]; Manipulate[ Show[ Graphics[{Line[{p1, p2}]}, PlotRange \[Rule] 3, Axes \[Rule] True], ParametricPlot[ {u[p1[[1]] + t (p2[[1]] - p1[[1]]), p1[[2]] + t (p2[[2]] - p1[[2]])], v[p1[[1]] + t (p2[[1]] - p1[[1]]), p1[[2]] + t (p2[[2]] - p1[[2]])]}, {t, 0, 1}, PlotRange \[Rule] 3]], {{p1, {0, 1}}, Locator}, {{p2, {1, 2}}, Locator}] 

(ugly, yes, but time cannot be fixed). Typical Output:

or

The idea is that you can change the line on the left side of the figures that you give (by clicking on the plot, which is a click on the Argand diagram ...) and see the corresponding images.

Answer 2

Depending on what you want to do with the views, it can sometimes be useful to render the Riemann surface in 3D. Here is the surface for w=sin(z) in 3D, neatly showing sections of branches and different branches (the same as the first acl plot, but in 3D).

 ParametricPlot3D[ Evaluate[{ Re@Sin [z], Im@Sin [z], y} /. z -> x + I y], {x, -2, 2}, {y, -2, 2}]