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    $\begingroup$ A plot of the function indicates that a continuous branch can be obtained by negating the function between 2Pi/3 and 4Pi/3. $\endgroup$ Commented Oct 16, 2013 at 16:09
  • $\begingroup$ Could you explain it (especially between $2\pi/3$ and $4\pi/3$) in detail? $\endgroup$ Commented Oct 16, 2013 at 18:39
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    $\begingroup$ I just looked at Plot[Re[I*Exp[I*t]/(4 Exp[I*t]^2 + 4 Exp[I*t] + 3)^(1/2)], {t, 0, 2 Pi}] and likewise for the imaginary part. They indicate jumps at the points I had stated, and visually it is clear that negating between those points will give a continuous branch. I realize this is not a proof, but it does indicate how you can proceed to get a numerical result. The two values are +-Pi*I, by the way. $\endgroup$ Commented Oct 16, 2013 at 23:15
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    $\begingroup$ I could be wrong, but how is this a Mathematica question? Isn't it a mathatics question? $\endgroup$ Commented Nov 26, 2013 at 22:07
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    $\begingroup$ As much as I recon this type of integrals were discussed in the book of Nikolos Muschelischwili "Some basic problems of the mathematical theory of elasticity". P. Noordhoff, Groningen 1953, where a general approach has been formulated, a rather easy one. $\endgroup$ Commented Nov 28, 2013 at 9:38