Residues of a quartic denominator in contour integration

by Mike Furson   Last Updated May 16, 2019 04:20 AM

So im working on an integral log^2(z^2)/(1+z^4) around upper half semicircle contour.

for singularities I get exp(ipi/4) and exp(3ipi/4) for the singularities in the contour

However, my answer is apparently wrong, because I get different answers for log^2((exp(ipi/4)^2) and log^2((exp(3ipi/4)^2)

are they supposed to be the same? if so, how/why?

the sum of denominator residues I know is right, but I only get the answer if both numerator residues equal -pi^2/4 which I dont know why its supposed to be like that.

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