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I have$$\int_{\gamma}\frac{z+1}{z(4z^2-1)^2}dz $$ where ${\gamma}$ is the positively oriented circle about i with radius 1/2.

$${\gamma}(t) = i +1/2e^{it}, 0 \leq t \leq 2 \pi$$.

I believe cauchy's integral formula does not apply in this case because the root +- 0.5 is not within the contour. So how does one go about solving this?

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The contour doesn't contain poles $z=0,\pm\dfrac12$, then the integral is zero by Cauchy's integral formula.

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