How to integrate $\int_{\gamma_1} \frac{dz}{z(z-i)}$ with $\gamma_1 = Re^{it}$, $R>1$?

I am stuck calculating the integral $$\int_{\gamma_1} \frac{dz}{z(z-i)}$$ over $\gamma_1 = Re^{it}, R>1$.

If I had to integrate over $\gamma_2 = re^{it}, r < 1$, I could just expand the integrand into a power series (using the geometric series) around $z=0$, but with $R>1$ this approach won't work. I quite frankly don't have any other idea on how to approach this.

Any hints?

2022-07-25 20:46:07
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