It says on Alfors, Complex Analysis page 73, chapter 2.3 that:
Suppose that an arc $\gamma$ with the equation $z = z(t), \alpha \leq t \leq \beta$, is contained ina region $\omega$, and let $f(z)$ be defined and continuous in the region. Then the equation $w = w(t) = f(z(t))$ defines an arch $\gamma '$in the $w$-plane which may be called the image of $\gamma$. Now, consider the case where $f(z)$ is analytic. If $z'(t)$ exists, $w'(t) = f'(z(t))z'(t)$. At the point $z_0 = z(t_0)$ and all derivative we are considering are nonzero, then $\gamma'$ has a tangent at $w_0 = f(z_0)$ and its direction is given by $\arg w'(t_0) = \arg f'(z_0) + \arg z'(t_0)$.
Then follows a part which I cannot understand:
"This relation asserts thay the angle between the directed tangents to $\gamma$ at $z_0$ and to $\gamma '$ at $w_0$ is equal to $\arg f'(z_0)$. It is hence independent of the curve $\gamma$.
I am struggling to understand this interpretation of the mathematical formula. I was wondering if anyone could give a mathematically rigorous description of what the author is saying.
Visually, you should think of $z'(t_0)$ as the tangent vector of the curve $z(t)$ at $t_0$. Similarly, $w'(t_0)$ is the tangent vector of $w(t)=f(z(t))$ at the point $z(t_0)$. However, from the analysis point of view, we can also regard these tangent vectors as complex numbers. The expression $\arg f'(z_0) = \arg w'(t_0)-\arg z'(t_0)$ then shows that $\arg f'(z_0)$ is the angle between the two vectors. (Note that $\arg z'(t_0)$ is the angle between the positive $x$-axis and the vector $z'(t_0)$; similar interpretation for $\arg w'(t_0)$.)
If we would take another curve $z_2(t)$ that goes through the same point $z_0$, then the angle between $z_2'(t_0)$ and $w_2'(t_0)$ would also be $\arg f'(z_0)$. Hence the value is independent of the curve we choose.