Question: I want to try plot a conformal map

Hello!

I just want to try plot a conformal map for better understanding how it works.

For example, I want to plot a conformal map, so I do this:

It works fine. But if I want to map more complecated region, I have lot's of problem. Could you help me please.

For example:

restart;
assume(y, real);
assume(x, real);
f := z -> I + z*exp(1/4*I*Pi);
w := f(x + y*I);
u := Re(w);
v := Im(w);
A := array(1 .. 2);
A[1] := plots:-conformal(z, z = 0 .. 1 + I/2, grid = [16, 16], numxy = [16, 16], scaling = constrained);
A[2] := plots:-conformal(f(z), z = 0 .. 1 + I/2, grid = [16, 16], numxy = [16, 16], scaling = constrained);
plots:-display(A);

how to map a triangle [0,0], [1,0], [0,i] or half-plane Im z> 1, excluding the circle | z - 2i | <= 1.

Thank you!