Does the Winding Number Determine Interior Points in Complex Domains?

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In summary, the conversation discusses a theorem that states that for a given open, connected, and bounded region D in the complex plane, the winding number of a point z is equal to 0 if and only if z is not in D. This theorem is proven using Cauchy's Integral Theorem and the concept of simply connected regions. The conversation also mentions the importance of Cauchy's Integral Formula and Theorem and residue theorems in this context.
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Poopsilon
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Homework Statement



Let D ⊂ C be open, connected, and bounded. Suppose the boundary of D consists of a finite number of piecewise differentiable simple closed curves: α0,...,αN, with α1,...,αN contained in the interior of α0. Suppose α0 is oriented in the positive direction and α1, . . . , αN oriented in the negative direction. Let α = α0 + · · · + αN . Show that for z ∈ ℂ\α, χ(α, z) = 0 if and only if z ∈ D.

Homework Equations



Well this formula is probably important: [itex] χ(α, z) = \frac{1}{2\pi i}\int_\alpha \frac{dζ}{ζ-z}[/itex].

I'm not sure what others might be important, maybe Cauchy's Integral Formula & Theorem, possibly some residue theorems..

The Attempt at a Solution



I mean χ(α, z) measures the winding number of a point z, so at first it seems that χ(α, z) = 0 when z is not in D, yet that's the opposite of what the theorem is saying for left to right.

So let's say I start with the right to left direction by assuming z ∈ D, then that means I need to prove that [itex]\int_\alpha \frac{dζ}{ζ-z} = 0[/itex].

Well by Cauchy's Integral Theorem, if I can prove that [itex]f(ζ) = \frac{1}{ζ-z}[/itex] is analytic in some simply connected region containing our curve α, then that would suffice.

Now I'm picturing D as some sort of disk like blob with some holes in it, where a0 is its outer boundary and a1 through aN are the boundaries of its holes. Thus since ℂ\α is open I will be able to sneak a disk around each of the holes which excludes z, wherever it might be in D, and thus [itex]f(ζ) = \frac{1}{ζ-z}[/itex] will be analytic on each one of those disks (since they are simply connected) and thus the integral [itex]\int \frac{dζ}{ζ-z} = 0[/itex] for the curves α1 through αN.

As for α0, I'm thinking that we could look at the region outside some disk that is just inside a0, and close enough to it to exclude z. And then from here I'm thinking that this region actually is simply connected through the point at infinity ( gosh I sure wish I knew more topology ). And then from that I can show that [itex]\int_{\alpha_0} \frac{dζ}{ζ-z} = 0[/itex].

But then this entire argument worries me, because for the other direction I assume z is not in D, then I can just stick a disk around all of D which excludes z and then again by Cauchy's Integral Theorem [itex]f(ζ) = \frac{1}{ζ-z}[/itex] is analytic in that disk and so its integral is zero for all the curves α0 through αN. So I think maybe this all spiel is wrong, maybe some of you could help me out, thanks.
 
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  • #2
If you imaging a simple ring shaped open region, where α0 is the outer circle, α1 is the inner circle, what you try to prove is certainly not true ...
 
  • #3
I'm sorry but I can't do anything with that comment.
 

FAQ: Does the Winding Number Determine Interior Points in Complex Domains?

1. What is the winding number iff statement?

The winding number iff statement is a theorem in mathematics that relates to the concept of winding numbers in complex analysis. It states that a point z is contained in the interior of a closed curve C if and only if the winding number of the curve with respect to z is non-zero.

2. How is the winding number calculated?

The winding number of a closed curve C with respect to a point z is calculated by counting the number of times the curve winds around the point in a counterclockwise direction. If the curve winds around the point clockwise, the winding number is negative.

3. What is the significance of the winding number?

The winding number is significant in complex analysis as it helps determine the behavior of complex functions. It is also used to classify the behavior of curves in the complex plane and can be used to identify singularities and poles of a function.

4. Can the winding number be fractional?

No, the winding number is always an integer value. This is because it represents the number of times a curve winds around a point in a counterclockwise direction, and this cannot be a fractional value.

5. How is the winding number iff statement used in real life?

The winding number iff statement has various applications in real life, such as in physics and engineering. It can be used to analyze the behavior of electric and magnetic fields around closed curves, and in fluid dynamics to study the flow of fluids around obstacles.

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