Suppose you have a unit circle in the complex plane [tex] e^{it}, -\infty \leq t \leq \infty[/tex]. The contour will wind around forever, so at all points in the contour, there are an infinite amount of possible winding numbers, although they are all multiples of 2pi with a well defined contour boundry, such as if t started at zero, they are not well defined if t starts at negative infinity, so this is what makes me think that they (the winding numbers) don't have to "step" by multiples of 2pi at each point in the contour, rather they can take all values in between also.(adsbygoogle = window.adsbygoogle || []).push({});

I was wondering if maybe we could exend another dimention to the complex plane to make it "the complex volume" with a real axis, an imaginary axis, and a winding number axis. so now this is kind of like a new set of numbers, w= a + bi + cw, where all numbers can have an intrinsic winding number of their own regardless of wether or not it is part of a contour... I was trying to think of how a sphere could be defined in this way, and I haven't got that far... Is there any way this can be done?

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# Complex analysis

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