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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.

enter image description here

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?

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  • $\begingroup$ hi,@Harry49 the question ask me to find $\int Rf(z)dz $ for each segment from A to D and use the result I found for $\int Rf(z)dz $ along each segment from A to D to find $\int f(z)dz $ (along the whole loop) $\endgroup$
    – kevin
    Commented Oct 25, 2017 at 9:58
  • $\begingroup$ @Harry49 btw thanks for your reply! $\endgroup$
    – kevin
    Commented Oct 25, 2017 at 10:00
  • $\begingroup$ It is possible to say Radius of A is zero due to Jordan's lemma?? $\endgroup$
    – kevin
    Commented Oct 26, 2017 at 10:35
  • $\begingroup$ Radius of A I mean the radius of big circle $\endgroup$
    – kevin
    Commented Oct 26, 2017 at 10:36
  • $\begingroup$ Therefore for part (A) is just zero? $\endgroup$
    – kevin
    Commented Oct 26, 2017 at 10:36

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