I am working with the function $f(z) = \frac{z+1}{z-1}$, for a complex variable $z$. I understood that for $z$ in the unit disc, i.e $\lvert z\rvert \le 1$, $\mathrm{Re}(f(z)) \le 0$.

What if $z$ is in a disc, say $D = \left\{ z \in \mathbb C: \, \lvert z\rvert \le a \right\}$ for a given constant $a \in \mathbb R_{+}$?

Note that the function is it's own inverse, if it matters.. I'm not very good with analysis and i have trouble visualizing functions from $\mathbb C$ to $\mathbb C$, since there are 4 dimensions...

I am working with the function $f(z) = \frac{z+1}{z-1}$, for a complex variable $z$. I understood that for $z$ in the unit disc, i.e $\lvert z\rvert \le 1$, $\mathrm{Re}(f(z)) \le 0$.

What if $z$ is in a disc, say $D = \left\{ z \in \mathbb C: \, \lvert z\rvert \le a \right\}$ for a given constant $a \in \mathbb R_{+}$?

Note that the function is its own inverse, if it matters …. I'm not very good with analysis and I have trouble visualizing functions from $\mathbb C$ to $\mathbb C$, since there are 4 dimensions ….

I am working with the function $f(z) = \frac{z+1}{z-1}$, for a complex variable $z$. I understood that for $z$ in the unit disc, i.e $\lvert z\rvert \le 1$, $Re(f(z)) \le 0$.

What if $z$ is in a disc, say $D = \left\{ z \in \mathbb C: \, \lvert z\rvert \le a \right\}$ for a given constant $a \in \mathbb R_{+}$?

Note that the function is it's own inverse, if it matters.. I'm not very good with analysis and i have trouble visualizing functions from $\mathbb C$ to $\mathbb C$, since there are 4 dimensions...

I am working with the function $f(z) = \frac{z+1}{z-1}$, for a complex variable $z$. I understood that for $z$ in the unit disc, i.e $\lvert z\rvert \le 1$, $\mathrm{Re}(f(z)) \le 0$.

What if $z$ is in a disc, say $D = \left\{ z \in \mathbb C: \, \lvert z\rvert \le a \right\}$ for a given constant $a \in \mathbb R_{+}$?

Note that the function is it's own inverse, if it matters.. I'm not very good with analysis and i have trouble visualizing functions from $\mathbb C$ to $\mathbb C$, since there are 4 dimensions...