there is this question which says an operation $z^{−1/2}$ has a branch function defined by $f(z) = r^{−1/2} e^{−iθ/2}$, where $z = re^{iθ}$ and $θ ∈ (−π,π)$. The branch cut lies along the negative real axis.
Consider the following loop
(A): anticlockwise arc of radius $R_A$, from just below the branch cut to just above.
(B): rightward line just above the branch cut.
(C): clockwise arc of radius $R_C$.
(D): leftward line just below the branch cut.
By parameterizing the contours, calculate $\int\Re f(z)\,\mathrm{d}z$ along each of the segments A, B, C, and D. Give your answers in terms of $R_A$ and $R_C$. Hence, find $\int f(z)\,\mathrm{d}z$ along the whole loop, and comment on how your result relates to Cauchy’s Integral Theorem.
This is the first time I am doing Cauchy integral theorem relating to branches, so for this case I'm not sure how should I start with. Any ideas?