\[ \textbf{Proposition 16.5} \text{ If the smooth loop }\gamma \text{ is monotonic about }p \text{, then  } wn(\gamma,p)=rot(\gamma) \]Generalize the above proposition by proving that the difference  \(wn( p) -rot(\gamma )\) is “in general” equal to the number of rays through \(p\) that are tangent to \(\gamma\), counted with appropriate signs. (“In general” refers to a transversality hypothesis.)

From geometry we know we can construct two rays tangent to a circle from any point outside the circle. Because the unit tangent vector of a loop is oriented, and we only consider the rays coinciding with the unit tangent vector,  we only consider one of the two tangent rays. The one coinciding with the clockwise or counter-clockwise orientation as the case may be.

Let \(\gamma:[0,1]\rightarrow \mathbb{C} \backslash \{0\} \) be a loop. Consider the set \( \omega=\{ \omega_1, \omega_2,.., \omega_{n-1}, \omega_{n} \} \) of inner loops of \(\gamma\) with  \(rot(\omega_k)=\pm 1\) and the set \(M=\{ m_1, m_2, \ldots, m_{r-1}, m_{r} \} \) of path components in order of non-decreasing winding number.  That is, \(wn(m_k) \le wn(m_{k+1})\). By the monotonicity condition on \(\gamma\), if   \( rot(\omega_{k})=1\) for some natural number \(k\) then  \( rot(\omega_k)=1\) for all \(0<k\leq n\). Similarly for  \( rot(\omega_{k})=-1\).

By theorem 8.7 we may replace \(\omega_k\) by circles for all \(0<k \leq n\) and connect them accordingly to make up \(\gamma\). Let \( p_0\in w_0\) be a point in the unbounded component, then \(wn(\gamma,p_0)=0\)  and it is outside the loops in \( \omega \) so we may construct one ray per inner loop, now circles, or \( n\) rays.

Now let \( p_{k} \in w_{k}\) . By construction, \(wn(p_{k},\gamma)=\sum_{s=0}^{r} rot(w_s)=r\), where \(p_{k} \in int(\omega_s)\) for all \(s\). Therefore, no tangent may be constructed from \(p_{k}\) to \(w_{s}\) for any \(0<s \leq r\). That leaves \(p_{k}\) in the exterior of a number of loops in \(\omega\), exactly \(rot(\gamma)-r\) loops. Since \(p_k\) is outside these loops, transformed into circles by homotopies, we can construct a ray to each circle. Hence, the number of rays from \(p_k\) to \(\gamma\) is \(rot(\gamma)-wn(p_s,\gamma)\). Or |\(rot(\gamma)-wn(p_s,\gamma)\)| when \(rot(w_k)=-1\) for some integer k.

I think this covers it. Let me know of any errors or suggestions.