Consider an anlytic curve $ C:z\left(t\right)=x\left(t\right)+iy\left(t\right) $.
Then by the book I'm reading, the angle between the curve and the real positive axis at a point $t_0$ , is given by the angle between the tangent line to $ C $ at $ t_0 $ , and the real positive axis. That is, the direction of the tangent line is $ \dot{z}\left(t_{0}\right)=x'\left(t_{0}\right)+iy'\left(t_{0}\right) $ So that the angle should be $ \text{Arg}\left(\dot{z}\left(t_{0}\right)\right) $.
Now, my lecturer gave another definition in class, I'd like to know what's the connection between those definitions. According to class, if $ C $ is analytic, then we can write $$ z\left(t\right)=z\left(t_{0}\right)+\frac{1}{n!}z^{(n_{0})}\left(t_{0}\right)\left(t-t_{0}\right)^{n_{0}}+O\left(t-t_{0}\right)^{n_{0}+1} $$
Where $n_0 $ is the first index such that the $n_0 $ derivative does not vanish. And then, the angle given by $ \text{Arg}\left(z^{(n_{0})}\left(t_{0}\right)\right) $
this gives different results because when $ \dot{z}\left(t_{0}\right) =0$ We'll get angle 0 (mod $2\pi$).
So, whats the realation between those two definitions? What's the accepted one?