I am trying to solve the following question: for each $a\in \mathbb{C}\setminus S^1$ calculate the degree of the map $$ \phi_a: S^1\to S^1; z\mapsto \frac{z-a}{|z-a|} $$ and deduce that $a,b\in \mathbb{C}\setminus S^1$ are in the same path component if and only if $\phi_a$ and $\phi_b$ are homotopic.
I am not sure how to start with the degree, should it be something like the highest integer less than $|a|$?
Consider the loop $\gamma\colon t\mapsto e^{2\pi i t}$ which winds once around $0$ when $t$ varies from $0$ to $1$. Under $\phi_a$ this becomes $$ \phi_a\circ\gamma\colon t\mapsto \frac{e^{2\pi i t} - a}{|e^{2\pi i t}-a|}. $$ The degree of $\phi_a$ is the winding number of $\phi_a\circ\gamma$ around $0$. Since $|e^{2\pi i t}-a|>0$ for $a\in\mathbb C\setminus S^1$, the denominator can't change the winding number (it's just a radial factor), so we can instead look at the winding number of $$ t \mapsto e^{2\pi t} - a $$ around $0$.
This loop describes a unit circle around $-a$ winded once. Can you see for which $a\in\mathbb C\setminus S^1$ this circle goes around $0$ once and for which it doesn't go around $0$ at all?