I am doing some plotting for my own interest of Mobius transforms, but my current system uses $u$ and $v$ axes where $u=u(x,y)$ and $v=v(x,y)$. I want to plot some Mobius transform f(z) as functions of $u(x,y)$ and $v(x,y)$, but I am not entirely certain how to do so.
I have attempted the typical approach of setting $z=x+iy$, which gives $f(z) = f(x, y)$ where $f(x,y) = \Re(f(x,y)) + i\Im(f(x,y))$.
I am guessing that I need to somehow represent $u(x,y) = \Re(f(x,y))$ and $v(x,y) = \Im(f(x,y))$. If this is correct, how would I do so?
My use case, for additional context, is plotting a small region (grid) of $x,y\in[-10,10]$ using $u(x,y)$ and $v(x,y)$. For my purposes, I require the transform to also be in this format, but this is not something I have ever needed to do before, and I am unsure as to the correct process. If you want to see more explicitly why, I am using this small visualisation I made a while ago, which shows exactly what I mean.
Edit: As an example of what I mean, suppose I have $f(z) = \frac{z-i}{z+i}$, and wanted to plot this as functions of $u(x,y)$ and $v(x,y)$ where $z = x + iy$.
So does that then simply imply that I need to substitute $z$ in $f(z)$ to get $f(x,y)$, and then equate $u(x,y)$ and $v(x,y)$ to the real and imaginary parts? Or is there more that I am missing?