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The video https://www.youtube.com/watch?v=wZvFKcQ_3Rc&t=8s mentioned something called the Index Theory. I can't find it on wikipedia. Where could I find more about the theory?

Here index is just the winding number of a curve in a vector field.

Also, the video mentioned that "In a vector field, every closed orbit must eclipse fixed points that have indices sum to 1".

My questions are: Is there some analogous results in complex analysis? Is there some connection between topology and complex integrals?

Since $\frac{1}{2\pi i}\int_\gamma \frac{1}{z-w}$ is the winding number of $\gamma$ around $w$. I guess many complex analysis results are consequences of applying topology to study differential equations. I would be grateful if someone could explain further the underlying topology in complex analysis.

HIH
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    Around 7:26 the author defines the index essentially by the line integral of the famous closed but not exact one form $$\boldsymbol{\omega}=\frac{g,df-f,dg}{f^2+g^2}$$ which is discussed here. There is an analogy to winding numbers in complex analysis but afaik index theory is built on generalizations of $\omega$ to higher dimensions. De Rham cohomology is the buzz word you should be looking for and this WP article shows quickly that even that is not everything. – Kurt G. Nov 14 '23 at 20:18

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