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I am going through the property (ii) of winding number from L.V.Ahlfors' book where I have failed to understand the reasoning "and if $\gamma$ does not meet the segment we must have

$$\int_{\gamma} \left ( \frac {1} {z-a} - \frac {1} {z-b} \right ) dz = 0;$$ hence..."

Would anybody please make it clear to me?

Thank you in advance.

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It's because, as Ahlfors explained, the function that we are integrating has a primitive, which is $\log\left(\frac{z-a}{z-b}\right)$. Therefore, and since $\gamma$ is a loop, the integral is $0$:

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  • $\begingroup$ Oh! I see. I am too much fool. Sorry for that. $\endgroup$ – Arnab Chatterjee. Nov 5 '17 at 14:47
  • $\begingroup$ @ArnabChatterjee. No, you were not a fool. If my answer was useful, perhaps that you could mark it as the accepted one. $\endgroup$ – José Carlos Santos Nov 5 '17 at 14:48
  • $\begingroup$ Oh! Sure. But I have to wait for five minutes to accept your answer. $\endgroup$ – Arnab Chatterjee. Nov 5 '17 at 14:49

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