Question

Complex analysis problem
Find \(\int_\gamma z^{-1/2} dz\) where \(\gamma \) is upper half of unit circle from 1 to -1
Please, help

Answer 1

i forget
don't you so something like put \(z(t)=e^{it}\) so \(z'(t)=ie^{it}\) ?

Answer 2

I have to use branch cut, my Prof said I can see z^(-1/2) as principal branch cut of z^(-1/2). I don't get what he meant

Answer 3

\(z^(-1/2) = e^{(-1/2) logz}\)

Answer 4

is upper half of unit circle from 1 to -1 means this right?

Answer 5

Yes

Answer 6

But when using branch cut, the log is undefined on negative of real part, we don't have -1 on the curve, how to argue?