Compute the winding number about the origin (TOPOLOGY) Watch

number23
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The winding number of \gamma about is given as: w(\gamma, 0)=\frac{arg(\gamma(1))-arg(\gamma(0))}{2\pi}

\gamma : [0,1] \rightarrow \mathbb{C}-\{0\} is a loop in \mathbb{C} not passing through the origin (so \gamma(0)=\gamma(1))

As \gamma is a loop arg(\gamma(1))-arg(\gamma(0)) is an integer multiple of 2\pi


Question 1: Compute the winding number of the loop \alpha(z)=4z^4+2z^2+1: S^1 \rightarrow \mathbb{C} - { } about the origin


Question 2: Compute the winding number of the polynomial loop \alpha(z)=6z^2+7z+2: S^1 \rightarrow \mathbb{C} - { } about the origin

The only way I can think of doing these is to prove the loops are homotopic to a simple loop, such as 4z^4 and 6z^2 respectively (not sure this is even correct) but I am unsure.

How would you go about these types of problems?
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Gregorius
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(Original post by number23)
The winding number of \gamma about is given as: w(\gamma, 0)=\frac{arg(\gamma(1))-arg(\gamma(0))}{2\pi}

\gamma : [0,1] \rightarrow \mathbb{C}-\{0\} is a loop in \mathbb{C} not passing through the origin (so \gamma(0)=\gamma(1))

As \gamma is a loop arg(\gamma(1))-arg(\gamma(0)) is an integer multiple of 2\pi


Question 1: Compute the winding number of the loop \alpha(z)=4z^4+2z^2+1: S^1 \rightarrow \mathbb{C} - { } about the origin


Question 2: Compute the winding number of the polynomial loop \alpha(z)=6z^2+7z+2: S^1 \rightarrow \mathbb{C} - { } about the origin

The only way I can think of doing these is to prove the loops are homotopic to a simple loop, such as 4z^4 and 6z^2 respectively (not sure this is even correct) but I am unsure.

How would you go about these types of problems?
I think that the point of a question like this is to show that the curves actually wind around the origin the number of times that you think they ought to!

So, if you take the first one, \alpha(z)=4z^4+2z^2+1, consider it in two parts: the 4 z^4 bit and the 2z^2+1 bit. Think of the result as the sum of a vector representing the first bit going out from the origin, with a vector representing the second bit added on to the end of the first. Now look at the modulus of the first vector in comparison with the second vector. Is it clear that the tip of the sum of the two vectors must have the same winding number as the first vector on its own?

The second one is more tricky as the length of the second component is sometimes greater than that of the first; see where this occurs and see what the argument of both vectors are at these points.

Addendum: If you know the argument principle from complex analysis, then you could apply that. This gives you that, if C is the unit circle in \mathbb{C}

\displaystyle \oint \frac{f'(z)}{f(z)} dz

is  2 \pi i times the winding number of f(C), where the integral is taken around C.
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