so if $j$ is odd, then at least one of these differences, say $$g_{\tau + 1} - g_{\tau}, \quad 0 \leq \tau < j,$$ is even, and the factor $W(g_{\tau}, g_{\tau + 1})$ in by is zero. Therefore, $u(g) = 0$, and this holds for all elements $u(g) \in $ so $\sum u(g) = 0$.

\[lem:tjeven\] If $j$ is even and $W$ is real, then for $n \geq N^*$ ($N^*$ as defined in ), $$T_j(n) {\equiv}T_j(n; iW) \text{ is real.} \label{eq:tevenreal}$$

Now the number $j = 2q$ of $W$-factors in the product is even for any $g \in {\mathbb{Z}}_+^{(j-1)}$ so $$p(g) = \prod_{t = 0}^{j-1} iW(g_t, g_{t + 1}) = (-1)^q \cdot \text{(real number)}$$ is real by or . and $$\begin{gathered}
u(g) = p(g) \cdot J(g), \text{ where} \label{eq:udecomp} \\ J(g) = \frac{1}{2\pi i} \int\limits_{{\partial (\mathcal{D}_n )}} F_g(z) \, dz \label{eq:jdef}, \text{ with} \\
F_g(z)  = (z - z_n) \cdot \frac{1}{(z - z_m)^2} \prod_{t = 1}^{j-1} \frac{1}{z - z_{g_t}}  \label{eq:fgzdef}\end{gathered}$$ For any $g$ this integral is real number \[see the next lemma\]. Therefore, $T_j(n)$ — as a sum of (absolutely) convergent series with real terms — is a real number.

This completes the proof of Proposition 4.1.

We will need 