Effectivisation of a string solution of the $2D$ Toda hierarchy and the Riemann theorem about complex domains

Let $0\in D_+$ be a connected domain with analytic boundary on the complex plane $\mathbb C$. Then according to the Riemann theorem there exists a function $w(z)=\frac{1}{r}z+\sum_{j=0}^\infty p_jz^{-j}$, mapping biholomorphically $D_-=\mathbb C\setminus D_+$ to the exterior of the unit disk $\{w\in\mathbb C\colon|w|>1\}$. From Wiegmann's and Zabrodin's rezults it follows that this function is described by the formula $log w=\log z-\partial_{t_0}(\frac{1}{2}\partial_{t_0}+\sum_{k\geq1}\frac{z^{-k}}{k}\partial_{t_k})v$, where $v=v(t_0,t_1,\bar t_1, t_2, \bar t_2,\dots)$ is a function of an infinite number of harmonic moments $t_i$ of the domain $D_-$. This function is independent from the domain and satisfies the dispersionless Hirota equation for the $2D$ Toda lattice hierarchy. In the paper we find recursion relations for coefficients of the Taylor series of $v$.