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The Index of a Curve is an Integer
Автор: Mike, the Mathematician
Загружено: 2025-04-15
Просмотров: 211
Описание:
We consider closed curves in the complex plane. We define the index of a curve about a point, p, not on the curve as the integral of dz/(z-p) over the curve (divided by 2(pi)i). Since we expect the antiderivative to be the log(z-p) = log|z-p| + i arg(z-p), we have an intuition that this will measure the number of times that the curve rotates around the point. We will prove that this quantity is in fact and integer and give examples which will make this proof more clear.
#mikethemathematician, #mikedabkowski, #profdabkowski, #complexanalysis, #indexofacurve
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