I must find the value of $\displaystyle \int _{C} z^{-1+i} dz$, where the integrand is such that $\displaystyle 0<\arg z<2\pi$ and $\displaystyle C$ is the positively oriented circle of radius 1.

My attempt : First I parametrize $\displaystyle C$ by $\displaystyle \gamma (t)=\cos t +i \sin t$.

Second, I try to find $\displaystyle f'(z)$. The only method I know is to use the definition : $\displaystyle f'(z)=\lim _{h \to 0}\frac{(z+h)^{-1+i}-z^{-1+i}}{h}$. I don't know how to calculate this limit... I'd like a hint.

Thanks!