4.I.4A

Complex Analysis | Part IB, 2005

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

(a) If γ1\gamma_{1} and γ2\gamma_{2} are closed paths not passing through zero, and if γ:[0,1]C\gamma:[0,1] \rightarrow \mathbf{C} is defined by γ(t)=γ1(t)γ2(t)\gamma(t)=\gamma_{1}(t) \gamma_{2}(t) for all tt, then

n(γ,0)=n(γ1,0)+n(γ2,0)n(\gamma, 0)=n\left(\gamma_{1}, 0\right)+n\left(\gamma_{2}, 0\right)

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

(c) If γ1\gamma_{1} and γ2\gamma_{2} are closed paths and aa is a complex number, not in the image of either path, such that

γ1(t)γ2(t)<γ1(t)a\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)-a\right|

for all tt, then n(γ1,a)=n(γ2,a)n\left(\gamma_{1}, a\right)=n\left(\gamma_{2}, a\right).

[You may wish here to consider the path defined by η(t)=1(γ1(t)γ2(t))/(γ1(t)a)\eta(t)=1-\left(\gamma_{1}(t)-\gamma_{2}(t)\right) /\left(\gamma_{1}(t)-a\right).]

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