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## Main Question or Discussion Point

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?

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?