2005-34.md

course	course_year	question_number	tags	title	year
Complex Analysis	IB	34	IB 2005 Complex Analysis	4.I.4A	2005

Let $\gamma:[0,1] \rightarrow \mathbf{C}$ be a closed path, where all paths are assumed to be piecewise continuously differentiable, and let $a$ be a complex number not in the image of $\gamma$. Write down an expression for the winding number $n(\gamma, a)$ in terms of a contour integral. From this characterization of the winding number, prove the following properties:

(a) If $\gamma_{1}$ and $\gamma_{2}$ are closed paths not passing through zero, and if $\gamma:[0,1] \rightarrow \mathbf{C}$ is defined by $\gamma(t)=\gamma_{1}(t) \gamma_{2}(t)$ for all $t$, then

$$n(\gamma, 0)=n\left(\gamma_{1}, 0\right)+n\left(\gamma_{2}, 0\right)$$

(b) If $\eta:[0,1] \rightarrow \mathbf{C}$ is a closed path whose image is contained in ${\operatorname{Re}(z)>0}$, then $n(\eta, 0)=0$.

(c) If $\gamma_{1}$ and $\gamma_{2}$ are closed paths and $a$ is a complex number, not in the image of either path, such that

$$\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)-a\right|$$

for all $t$, then $n\left(\gamma_{1}, a\right)=n\left(\gamma_{2}, a\right)$.

[You may wish here to consider the path defined by $\eta(t)=1-\left(\gamma_{1}(t)-\gamma_{2}(t)\right) /\left(\gamma_{1}(t)-a\right)$.]