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In this answer is it mentioned that

$$\frac{1}{(u+z)^2}\in\Theta(1/N^2),$$

I know that $$\frac{1}{2\pi i}\int_C\frac{1}{z^2}dz=0$$ where $C$ is any circle centered at the origin. But what about $$\frac{1}{2\pi i}\int_C\frac{1}{(u+z)^2}dz?$$ Can we make a change of variables to get that this integral is also zero?

Twnk
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    Please specify how $u,z,N$ are related. Say $z$ is on a certain contour defined in terms $N$, and we are interested in what happens when $N \to \infty$. – GEdgar Feb 22 '24 at 03:07
  • For the unrelated "I know" part. The function $z \mapsto 1/(u+z)^2$ has meromorphic antiderivative $-1/(u+z)$, so integrating on a closed curve (that does not pass through $-u$) is certainly $0$. – GEdgar Feb 22 '24 at 03:14

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