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Any ideas on how to approach this kind of question?? What does the question means I for each of the branches?? enter image description here

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With the branch cut along the positive real axis, we have

$$z^{1/5}=R^{1/5}e^{i\phi/5}e^{i2n\pi/5}$$

for $n=0,1,2,3,4$. The index $n$ defines the branches.

Hence, we have

$$I_n=-R^{6/5}e^{i2n\pi/5}\int_0^{2\pi} e^{i6\phi/5}\,d\phi$$

Can you finish now?

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  • $\begingroup$ hi @Mark Viola , thanks for helping...can I say that even though the diagram drew the circle in the sense it passes through the branch cut but actually it doesn't right?? $\endgroup$
    – kevin
    Commented Nov 16, 2017 at 16:13
  • $\begingroup$ Also I don't have to solve the contour like the one shown in the link??? upload.wikimedia.org/wikipedia/commons/thumb/8/8d/… $\endgroup$
    – kevin
    Commented Nov 16, 2017 at 16:14
  • $\begingroup$ You're welcome. My pleasure. $\endgroup$
    – Mark Viola
    Commented Nov 16, 2017 at 17:26
  • $\begingroup$ The contour begins at $\pi =2\pi ^-$ and ends at $\phi =0^+$. $\endgroup$
    – Mark Viola
    Commented Nov 16, 2017 at 17:27
  • $\begingroup$ I'm not sure as to your meaning of "solve the contour like the one shown in the link." $\endgroup$
    – Mark Viola
    Commented Nov 16, 2017 at 17:29

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