Exercise 8.23

Question

Exercise 23: 
    Let $\gamma$ be a continuously differentiable {\it closed} curve in the complex plane, with parameter interval $[a,b]$, and assume that $\gamma(t)
e 0$ for every $t\in[a,b]$. Define
    the {\it index} of $\gamma$ to be $$\Ind(\gamma)=\frac{1}{2\pi i}\int_a^b\frac{\gamma'(t)}{\gamma(t)}\,dt.$$ Prove that $\Ind(\gamma)$ is always an integer.

Compute $\Ind(\gamma)$ when $\gamma(t)=e^{int}$, $a=0$, $b=2\pi$.

Explain why $\Ind(\gamma)$ is often called the {\it winding number} of $\gamma$ around 0.

analambanomenos · Accepted Answer

Following the hint, for $x\in[a,b]$ define $$\varphi(x)=\int_a^x\frac{\gamma'(t)}{\gamma(t)}\,dt.$$ Then $\varphi(a)=0$, and by Theorem 6.20, since $\gamma'/\gamma$ is continuous on $[a,b]$,
    we have $\varphi'=\gamma'/\gamma$ on $[a,b]$. Let $f(x)=\gamma(x)e^{\varphi(x)}$, $x\in[a,b]$. Then $$f'(x)=\big(\gamma'(x)-\varphi'(x)\gamma(x)\big)e^{\varphi(x)}=0,$$ so $f$ is constant on
    $[a,b]$, equal to $f(a)=ae^{\varphi(a)}=a$. Hence $f(b)=be^{\varphi(b)}=a=b$, or $$e^{\varphi(b)}=e^{2\pi i\Ind(\gamma)}=1$$ so that $2\pi i\Ind(\gamma)=2\pi in$ for some integer $n$. Hence
    $\Ind(\gamma)=n$ is an integer.

For $\gamma(t)=e^{int}$, $a=0$, $b=2\pi$, we have $$\Ind(\gamma)=\frac{1}{2\pi i}\int_0^{2\pi}\frac{ine^{int}}{e^{int}}\,dt=\frac{2\pi in}{2\pi i}=n.$$ For these $\gamma$, $\Ind(\gamma)$
    measures the number of times $\gamma$ winds counter-clockwise around 0, with a negative value indicating that $\gamma$ winds around 0 clockwise $-\Ind(\gamma)$ times. This is true in a general
    sense for arbitrary $\gamma$, which is shown in most introductory Complex Analysis courses. Also, it can be shown that two such curves have the same index if and only if one can be
    ``continuously deformed'' to the other through an intermediate set of such curves $\gamma_t$.

Exercise 8.23

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Exercise 8.23

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