Complex Analysis: Defining Complex Volume & Sphere w/ Winding Number

[Jonny_trigonometry]

Suppose you have a unit circle in the complex plane $$e^{it}, -\infty \leq t \leq \infty$$. 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 boundary, 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.

I was wondering if maybe we could exend another dimension 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?

 

[matt grime]

You can certainly form the sets CxR or CxZ, but what you want to do with them is beyond me, since complex numbers do not have winding numbers.

 

[waht]

You could define a 3rd axis with a choice of real or complex number. It would be interesting how this works out in spherical or cylindrical coordinate systems.

I'm just guessing, maybe a surface integral of a closed surface would equal 0, (analogous to Cauchy's theorem) and you could do stuff with residues, that would be neat.

 

[waht]

You could define a 3rd axis with a choice of real or complex number. It would be interesting how this works out in spherical or cylindrical coordinate systems.

I'm just guessing, maybe a surface integral of a closed surface would equal 0, (analogous to Cauchy's theorem) and you could do stuff with residues, that would be neat.

 

[waht]

You could define a 3rd axis with a choice of real or complex number. It would be interesting how this works out in spherical or cylindrical coordinate systems.

I'm just guessing, maybe a surface integral of a closed surface would equal 0, (analogous to Cauchy's theorem) and you could do stuff with residues, that would be neat.

 

[waht]

oops sorry, had some connection problem.

 

[Jonny_trigonometry]

You can certainly form the sets CxR or CxZ, but what you want to do with them is beyond me, since complex numbers do not have winding numbers.

ya, i don't think this would work... but I'm beckoned by the idea that (at least for a contour) the way to distinguish how many times it was wound around a point is by simply "layering" the contour as if it's going up a spiral staircase. The point a + bi with winding # 2.3 is not the same as the point a + bi with winding # 2PI + 2.3... you know?

 

[matt grime]

You can certianly consider CxC too and look at it with holomorphic stuff in mind but it still has nottihng to do with the intrinsic winding number of a point since that makes no sense.

 

[matt grime]

ya, i don't think this would work... but I'm beckoned by the idea that (at least for a contour) the way to distinguish how many times it was wound around a point is by simply "layering" the contour as if it's going up a spiral staircase. The point a + bi with winding # 2.3 is not the same as the point a + bi with winding # 2PI + 2.3... you know?

in that case yo'u're verging towards riemann surfaces, which roughly speaking turn multiple valued tihngs like log or sqrt into single valued functions by gluing together copies of parts of C, but there is a different one for each function.

 

[Jonny_trigonometry]

You could define a 3rd axis with a choice of real or complex number. It would be interesting how this works out in spherical or cylindrical coordinate systems.
I'm just guessing, maybe a surface integral of a closed surface would equal 0, (analogous to Cauchy's theorem) and you could do stuff with residues, that would be neat.

ya i thought for a while about what it would be like to add a real axis perp to the complex plane, and that is basically like taking a bunch of complex planes and stacking them on top of each other. All cross products would stay within the plane of the two original vecotrs. The cauchy theorem could be used on arbitrary planes within this volume and stuff. It would be neat to explore all the implications, maybe it would be useful I don't know.

 

[Jonny_trigonometry]

You can certianly consider CxC too and look at it with holomorphic stuff in mind but it still has nottihng to do with the intrinsic winding number of a point since that makes no sense.

ya it doesn't make any sense, because it's based on the contour that the point is part of. It's like trying to define another dimension for all trig functions to distinguish between periods. For the winding number of an intrinsic point, it is always a function w/respect to some other point... What if that other point was dynamically changing? What would happen if you were to continuously compute the winding number for each point on a contour based on a different contour, say, the derivative of the first contour?

 

[Jonny_trigonometry]

in that case yo'u're verging towards riemann surfaces, which roughly speaking turn multiple valued tihngs like log or sqrt into single valued functions by gluing together copies of parts of C, but there is a different one for each function.

I'll check that stuff out... I just finnished complex analysis this semester, so I haven't really got into riemann surfaces and stuff yet, although they sound very cool.