No deep theorems required here. Just parameterize the curve and integrate.
On the circle |z|= 1, so and . Of course to cover the entire circle must go from 0 to . The integral is
On the circle |z-1|= 1, so and .
If .
On .
, C is along the circle |z|=1
and
another case, C along the circle |z-1|=1.
. C is |z|=1, and |z-1|=1
i think the first part is the same for both cases which is 2pi*i.
and the second part is 0 for both cases by Cauchy Integral Theorem.
But i am not sure if that can be applied. Because sqrt z is only analytic on C\ { x axis } . Therefore, out circles are not connected?
No deep theorems required here. Just parameterize the curve and integrate.
On the circle |z|= 1, so and . Of course to cover the entire circle must go from 0 to . The integral is
On the circle |z-1|= 1, so and .
If .
On .