Poles vs branch points

1. Nov 16, 2015

LagrangeEuler

I am a bit confused. Why $\frac{1}{\sqrt{z}}$ is branch point and $\frac{1}{z}$ is pole. And why we cannot use Cauchy integral theorem when we have branch point? Why we need to cut off branch point when we integrate? Thanks a lot for the answer.

2. Nov 16, 2015

mathman

$\frac{1}{\sqrt{z}}$ is double valued. Riemann surface has two sheets - branch cut needed.

3. Nov 17, 2015

LagrangeEuler

Yes I now that the problem is because is double valued. However I am not sure what is really happens. For example taking the circle around the point $z=0$ problem occurs. Could you explain where is the problem?