How to connect the winding number's definition to geometrical intuition?

In summary, according to the geometrical intuition, the winding number of a closed curve gamma is 1 if the point lies inside the curve, and 0 if the point is outside the curve. The definition of winding number of a closed curve gamma with respect to a is n(\gamma ,a) = \frac{1}{{2\pi i}}\int\limits_\gamma {\frac{{dz}}{{z - a}}} . The textbook of Lars Ahlfors says that we can write \int\limits_\gamma {\frac{{dz}}{{z - a}}} = \int\limits_\gamma {d\log (z - a) = \int\
  • #1
kakarotyjn
98
0
We know the winding number as a curve winds round a point.If a point is inside a closed curve,then the winding number is 1,this is the geometrical intuition.

The definition of winding number of closed curve gamma with respect to a is [tex]n(\gamma ,a) = \frac{1}{{2\pi i}}\int\limits_\gamma {\frac{{dz}}{{z - a}}} [/tex].

The textbook of Lars Ahlfors said we can write [tex]\int\limits_\gamma {\frac{{dz}}{{z - a}}} = \int\limits_\gamma {d\log (z - a) = \int\limits_\gamma {d\log |(z - a)|} } + i\int\limits_\gamma {d\arg (z - a)} [/tex],when z describes a closed curve,log|z-a| returns to its initial value and arg(z-a) increase or decreases by a mutiple of 2pi.This would seem to imply the lemma,but more careful thought shows that the reasoning is of no value unless we define arg(z-a) in a unique way.

I don't understand this scentence,if arg(z-a) is not define in a unique way,and how does it be defined in a not unique way,the number is not multiple of 2pi?
 
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  • #2
kakarotyjn said:
I don't understand this scentence,if arg(z-a) is not define in a unique way,and how does it be defined in a not unique way,the number is not multiple of 2pi?

Hi kakarotyjn!:smile:

We don't need arg(z-a) to be uniquely defined (which, as you say, it isn't),

we only need d(arg(z-a)) to be uniquely defined (which it is :wink:).
 
  • #3
Hi tiny-tim!Thanks for your reply!

Then how can we make d(arg(z-a)) uniquely defined?I'm really confused of this.
Can we define it in the complement of z>0?
 
  • #4
Hi kakarotyjn! :smile:

(just got up :zzz: …)
kakarotyjn said:
Then how can we make d(arg(z-a)) uniquely defined?I'm really confused of this.
Can we define it in the complement of z>0?

(i don't understand what you mean by "complement of z>0" :confused:)

arg has an ambiguity of ±2πn, but d(arg) doesn't, because we can assume that the same value of n is used …

so long as γ does not go through the point a, a small change in z makes a small change in arg(z-a) … ie, very nearly 0, not very nearly ±2πn. :wink:
 
  • #5
HiHi,tiny-tim:tongue2:(Just got up:rofl:)


the complement of z>0 means the complement of positive real axis.

and when they make some functions single value,for example log(z) ,the choose complement of negative real axis as the principle value brance,I'm not clear of that:confused:

Well,then why Lar Ahlfors said :This would seem to imply the lemma,but more careful thought shows that the reasoning is of no value unless we define arg(z-a) in a unique way.

Or why can't we think about the definition of winding number and just prove the formula of winding number is the winding number?Just as what he said :z describes a closed curve,log|z-a| returns to its initial value and arg(z-a) increase or decreases by a mutiple of 2pi.

Sorry for my poor English,and thanks for your help!:smile:
 
  • #6
Hi kakarotyjn! Good afternoon from London! :smile:

I think the answer is that, although arg(z-a) is multi-valued, this is a curve, and so is a function z(t) for some parameter t, and we can define arg(z(t)-a) to be single-valued (which is most easily done by simply requiring arg(z(t)-a) to be a continuous function of t). :smile:

(but using the complement of the positive real axis, or of any other line, won't work, because any closed curve will intersect that line … but it doesn't matter, because our single-value requirement is not for all z, or indeed for any open set, but only for the closed image z(R) :wink:)
 
  • #7
Yar!

That is to say,if we define arg(z-a) in a unique way,then the discussion is consistent.

Really thank you for your patience:smile:

Well,I have to go bed.

Good night from Weihai,haha!
 

1. What is the winding number's definition?

The winding number is a mathematical concept that describes the number of times a curve winds around a fixed point in a given direction.

2. How can the winding number be connected to geometrical intuition?

The winding number can be connected to geometrical intuition by visualizing the curve as a path taken by a point moving around the fixed point. The winding number then represents the number of revolutions made by the point around the fixed point.

3. What is the significance of the winding number in geometry?

The winding number has important applications in geometry, as it can be used to classify and distinguish between different types of curves. It is also used in complex analysis to study the behavior of complex functions.

4. How is the winding number calculated?

The winding number is calculated by counting the number of times the curve crosses a line drawn from the fixed point in the given direction. Each time the curve crosses the line, the winding number increases by 1. If the curve crosses the line in the opposite direction, the winding number decreases by 1.

5. Can the winding number be negative?

Yes, the winding number can be negative if the curve winds around the fixed point in a clockwise direction. This is because the winding number is based on the direction of traversal, rather than the actual number of revolutions made by the curve.

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