Center and angle of complex function

Question

Does a complex function of type $f(z)=az+b $ always have a center and angle (of rotation) or only when $b=0$ since $b\neq0$ represents a translation?

Answer 1

The function $f(z) = az+b$ can be rewritten as $$f(z) = a(z-c)+c$$ ($b=c-ac$, so $c = \frac b{1-a}$) iff $a\ne 1$. This, in turn is precisely a rotation by $\mathrm{arg}(a)$ and dilation by $|a|$ with respect to the center $c = \frac b{1-a}$. So the answer is yes, as long as $a\ne 1$ (or $b=0$ and $a=1$, i.e. $f(z)=z$ with "center" anywhere and "angle" $0$).
If $a = 1$, $f$ reduces to a translation wich doesn't have a sensible "center".