Let $C$ be a smooth projective curve defined over $\mathbb{Q}$. The regulator is a map
$$
reg: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}).
$$$$
\operatorname{reg}: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}).
$$ Here $K_2(C)_{\mathbb{Q}}$ is the rational second K-group tensor $\mathbb{Q}$ which which admits the description:
$$
\ker(K_2(\mathbb{Q}(C)) \longrightarrow \bigoplus_{x \in C(\overline{\mathbb{Q}})} \overline{\mathbb{Q}}^\times) \otimes_{\mathbb{Z}} \mathbb{Q}
$$ with$$
\ker\left(K_2(\mathbb{Q}(C)) \longrightarrow \bigoplus_{x \in C(\overline{\mathbb{Q}})} \overline{\mathbb{Q}}^\times\right) \otimes_{\mathbb{Z}} \mathbb{Q}
$$ where $K_2(\mathbb{Q}(C))$ is generated by symbols $\{f, g\}$ and the mapsmap is given by the tame symbols
$$
T_x(\{f, g\})=(-1)^{ord_x(f)ord_x(g)} [f^{ord_x g}/g^{ord_x f}](x).
$$
Since $H^1(C(\mathbb{C}), \mathbb{Q})$ is the dual of $H_1(C(\mathbb{C}), \mathbb{Q})$, to define the regulator of a symbol $\{f, g\}$, one needs to attach a real number $$ reg(\{f, g\})(\gamma) $$$$ \operatorname{reg}(\{f, g\})(\gamma) $$ to each $\gamma \in H_1(C(\mathbb{C}), \mathbb{Q})$.
To do so, one first starts defining $$ \omega(f, g)=\log |f|d arg(g)-\log |g| d arg(f) $$$$ \omega(f, g)=\log |f|\operatorname{d} \arg(g)-\log |g| \operatorname{d} \arg(f) $$ which is a closed real analytic 1-form on the complement of $S=div(f) \cup div(g)$$S=\operatorname{div}(f) \cup \operatorname{div}(g)$ on the Riemann surface $C(\mathbb{C})$. We first define the real number $$ \frac{1}{2\pi i} \int_{\gamma} \omega(f, g) $$ for $\gamma$ a cycle on $U=X-S$ (Itand notice that it actually only depends on the cohomology class of $\gamma$). Then we would like to say that this definition remains valid also for $\gamma$ in the whole of $X$. Since the cohomologies of $U$ and $X$ are related by residuesresidue maps $$ 0 \to H^1(X) \to H^1(U) \to \bigoplus_{s \in S} \mathbb{R} \to \mathbb{R} \to 0 $$ one needs to show that the residue vanishes at all points of $S$. What I have read is that this residue is given by $$ log | T_s(\{f, g\})| $$$$ \log | T_s(\{f, g\})| $$ (so since the tame symbol is torsion equal to zero) but I don't understand how to prove this. I guess I should take a small circle $C_s$ around a point in $S$ and show that $$ \frac{1}{2\pi} \int_{C_s} \omega(f, g)=\log |T_s(\{f, g\})|. $$ Can anybody help me proving this? This should be some kind of Cauchy theorem but since the integrand is not holomorphic I am a bit lost...
