Why is ##\frac{1}{\sqrt{z}}## a branch point and ##\frac{1}{z}## a pole?

[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.

 

[mathman]

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

 

[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?