Definition

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Let   be a cycle in  , and let   be a point that   does not intersect. Then

 

is called the *winding number* of   around  .

Motivation

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First, consider the case where   consists of a single closed curve. Then   is homologous in   to an  -fold (for some  ) traversed circle   around   with  . Now,

 

Thus, this integral counts how many times the curve   winds around the point  .

Task

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Let the closed integration path   be defined as:

 

1. Plot the trace of the integration path.

2. Determine the winding number  .

3. Determine the winding number  .

4. Determine the winding number  .

Additivity of the Integral

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For a cycle   with closed  , due to the additivity of the integral, we have

 

Thus, the winding number also counts how many times the point   is encircled.

Length of the Cycle

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For a cycle   with closed  , the length of the cycle is defined additively over the lengths of the individual integration paths:

 

See Also

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Translation and Version Control

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https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl

  • Date: 12/17/2024