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The following problem is part of the "easy computation" left to the reader in Arakelov´s article Intersection Theory of Divisors on an Arithmetic Surface, it is in the proof of Proposition 1.1.

In the setup $X$ is a Riemann Surface, $P$ is a point on it, and $\varphi_P(z)$ is a function which looks like $|z-P|\cdot u(z)$ in a neighborhood of $P$, with $u(z)$ meromorphic without poles or zeroes around $P$. Moreover $U_P$ is a small ball around $P$.

The computation should give $\int_{\partial U_P} \frac{\partial}{\partial n}log(\varphi_p(z))=-2\pi$. Here $\frac{\partial}{\partial n}$ is the normal derivative obtained applying Green´s Second Identity.

Applying the Residue Theorem one gets that the left hand side is equal to $2\pi i \:Res(\frac{\partial}{\partial n} log(\varphi_p(z)),P)$. Here I thought that, since the complex derivative is independent from the direction, then $\frac{\partial}{\partial n} log(\varphi_p(z))$ is the logarithmic derivative $\frac{\varphi_P(z)´}{\varphi_P(z)}$, and its residue is equal to $-1$ by definition of $\varphi_P(z)$.

Finally my computation gives $-2\pi i$ instead of $-2\pi$. The discrepance could be clearly a typo of the article (also because for the purposes of the proof the term $i$ doesn´t play any role). But it seems to me that my computation is messy, in specific when I handle with the normal derivative. Can please someone clear it up?

Thank you in advance for your time!

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