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I'm trying to integrate $\frac{1}{z^7+1}$ around the circle of radius 2, and was wondering how I express this in terms of partial fractions, hopefully as linear factors so as to apply the Cauchy integral formula. Is this possible? What is the general procedure for partial fractions when it comes to complex numbers? Thanks in advance!

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If $n$ is odd as in your example, then $z^n+1=0$ iff $(-z)^n=1$, hence in this case the denominator is $-\prod_{k=0}^{n-1}(z-\zeta^k)$, where $\zeta$ is a primitive $n$th root of unity. With a product of linear factors you can proceed with partial fractions just as in the real case.

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