Command in Maple

Question

I am looking for a possibility to draw the image of the two sets $$\{z\in\mathbb{C}:\text{Im}(z)>0,\text{Re}(z)=1/2\}$$ and $$\{z\in\mathbb{C}:\text{Im}(z)>0,|z|=1\}$$ under the map $z\mapsto\frac{z-i}{z+i}$ with Maple. It would be nice if this lines could be in the same plot.

I have read that the command conformal can help here?

I am looking forward to your answers. Thank you!

Answer 1

Here is a code that you can copy/paste.

with(plots):
A := conformal(z, z = .5 .. .5+1.*I, color = [green, red]):
B := conformal((z-I)/(z+I), z = .5 .. .5+10*I, numxy = [200, 200], color = [green, red]): 
C := complexplot(exp(I*x), x = 0 .. Pi):
E := complexplot((exp(I*x)-I)/(exp(I*x)+I), x = 0 .. Pi):
display(A, C);
display(B, E);

The command "with(plots)" loads the necessary package to use the next commands. "conformal(f(z),z=a..b)" produces the image of a grid from the point $a$ to the point $b$ on the complex plane. With "color=[X,Y]", we can define the color of the grid and "numxy" determine the number of sample for every line of the grid. Try modifying the values and see what it does.

"complexplot" is a command for plotting complex values. It is used like a normal "plot" here but takes complex values instead.

Finally "display" is used to combine multiple plots. The first one here displays the graph of the sets and the second one the image of those sets under the function $z\mapsto\frac{z-i}{z+i}$.

Note that I could have used "complexplot" only since the set you are working on are curves and that "conformal" produces grids. But I think it is a good thing that you can see what both commands do.