Connection Equation on the Riemann Surface

Abstract

Here we obtain the third equation for the Cauchy data using the identity (8.4). One of the central ideas of our method is that this identity also holds in the complex region, in the tube domain ${\mathbb C} K^*\subset {\mathbb C}^2$ defined by

$$\displaystyle \begin{aligned} \mathbb{C} K^*:=\{z\in\mathbb{C}^{2}: {{\mathrm{Im}\,} }z_1>0, {{\mathrm{Im}\,} }z_2>0\}. \end{aligned}$$