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http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-04Fall-2003/FACBFB25-64E5-4AA8-8868-F623EDA94CE8/0/assignment3.pdf [Broken]

This link on problem 4 is troubling me. What I want to do is evaluate it, but I first need to find the arguments on the segments above and below the cut. To do this I started off by factoring x^-2/3 out of the integrand, to get x^-2/3(1-x)^-1/3. After this, I restrict the argument of x in between pi and -pi, and for 1-x between 0 and 2pi. After doing out the algebra we find that the cut line is exactly between 0 and 3 on the real axis. Defining them in their respective local polar coordinates, we get on the top to be -2pi/3 and on the bottom zero for each argument. I know this is where I messed up because after factoring out the exp(-2pi/3) from the segment on the top, and calling the contour [1-exp(-2pi/3)]I and applying the residue theorem, i get the answer is imaginary! This has to be wrong, but why is it that my arguments are wrong, Any help would be GREATLY appreciated, so if you know anything about this stuff, then please help!

This link on problem 4 is troubling me. What I want to do is evaluate it, but I first need to find the arguments on the segments above and below the cut. To do this I started off by factoring x^-2/3 out of the integrand, to get x^-2/3(1-x)^-1/3. After this, I restrict the argument of x in between pi and -pi, and for 1-x between 0 and 2pi. After doing out the algebra we find that the cut line is exactly between 0 and 3 on the real axis. Defining them in their respective local polar coordinates, we get on the top to be -2pi/3 and on the bottom zero for each argument. I know this is where I messed up because after factoring out the exp(-2pi/3) from the segment on the top, and calling the contour [1-exp(-2pi/3)]I and applying the residue theorem, i get the answer is imaginary! This has to be wrong, but why is it that my arguments are wrong, Any help would be GREATLY appreciated, so if you know anything about this stuff, then please help!

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