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Need some help with computing index number (winding number). I need to compute the index number: $n(\gamma,0)$ where $\gamma(t)=\cos t + 3 i \sin t$ for $0 \leq t \leq 4\pi$

I tried to use parametrization, but the integral seem quite ugly. By the use of computer the answer is 0, but it can't be because the index counts the number of circuit on this ellipse, it should be 2 as i see it.

Maybe you have a way to compute it?

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    @Matt: You deleted the factor of $4$; I think that was included by intention.2011-11-22
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    @joriki: Unintended. Thanks, joriki.2011-11-22

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Clearly the winding number is $2$, because $2\pi$ worth of $t$ is one circuit around the origin, and there are two of them. This argument ought to be sufficient everywhere except in specialized classroom situations where you're required to use some particular method to determine the winding number. Since you're not specifying a method I assume this is not the case for you.

If you do want some kind of "more rigorous" argument, I recommend something like:

Let's construct a continuous argument function $f(t)$ for $\gamma$. Since $\gamma(0)=1$, let's set $f(0)=0$.

For $0\le t\le \pi$ the imaginary part of $\gamma(t)$ is nonnegative, so let's make $f(t)\in[0,\pi]$ in this interval. This leads, in some way we don't need to care about, to $f(\pi)=\pi$.

For $\pi\le t\le 2\pi$ the imaginary part of $\gamma(t)$ is nonpositive, so let's make $f(t)\in[\pi,2\pi]$ in this interval. This leads, in some way we don't need to care about, to $f(2\pi)=2\pi$.

For $2\pi\le t\le 3\pi$ the imaginary part of $\gamma(t)$ is nonnegative, so let's make $f(t)\in[2\pi,3\pi]$ in this interval. This leads, in some way we don't need to care about, to $f(3\pi)=3\pi$.

... and so forth ...

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    At a given $t$ the argument is $\tan^{-1}(3\tan t)$, which has derivative ${1 \over 1 + 9\tan^2 (t)}(3\sec^2(t)) \geq 0$. So the argument is increasing. It is zero at multiples of $2\pi$ so the curve wraps around twice.2011-11-22
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    I'll be very glad if you can explain it again. I didn't so much understand...Thanks.2011-11-22
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The winding number $n(\gamma, a)$ is defined by $$n(\gamma, a) = {1 \over 2\pi i} \int_{\gamma} {1 \over z - a}\,dz$$ So you are looking for $${1 \over 2\pi i} \int_{\gamma} {1 \over z}\,dz$$ The Cauchy integral formula will give that the integral over each circuit of this is $1$, so the overall answer is $2$.

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    Doesn't the Cauchy integral formula suppose that we _already_ know that $\gamma$ winds around the origin the appropriate number of times?2011-11-22
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    I think you can assume this for the ellipse. If not, scale the ellipse by 3 in the vertical direction. Surely you can assume it for a circle centered at the origin.2011-11-22
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    The point if that once you know how many times $\gamma$ winds around the origin, _that's the winding number right there_. The detour via the integral formula adds no value to the computation.2011-11-22
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    The definition of $n(\gamma,0)$ is the integral, not "how many times $\gamma$ wraps around the origin". So I am interpreting the question as asking the student to interpret it in terms of the Cauchy integral formula and correctly figuring out what the curve is, not asking in detail why an ellipse wraps around the origin once. The latter follows by scaling the circle anyhow.2011-11-22
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    Before you're allowed to use Cauchy's integral formula, you need to prove that its conditions are satisfied, among which is that _the curve must have winding number 1_. Once you have done this, you have already solved the original problem. There's nothing left to use the integral formula for.2011-11-22
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    What I'm saying is that the problem is to 1) recognize that the Cauchy integral formula applies here and 2) recognize that every $2\pi$ the ellipse wraps around once. I don't think they're being asked to rigorously prove the ellipse wraps around once. At least that's not how most complex analysis courses go. I can't know for sure in this case not being there myself.2011-11-22
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    Thanks about the answer. It seems that the curve is not important in this qeustion. If we use Caushy we can say that the index number n(γ,0) for every curve is same. This situation is acceptable? I think that there is a trick on the question. Don't you think?2011-11-22