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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**