Integrable systems and effectivisation of Riemann theorem about domaims of the complex plane

Abstract

Consider a closed analytic curve $\gamma$ in the complex plane and denote by > $D_+$ and $D_-$ the interior and exterior domains with respect to the curve. The point $z=0$ is assumed to be in $D_+$. Then according to Riemann theorem there exists a function $w(z)=\frac 1r z+\sum_{j=0}^\infty p_j z^{-j}$, mapping $D_-$ to the exterior of the unit disk $\{w\in C|| w | >1\}$. It is follow from [arXiv : hep-th /0005259] that this function is described by formula $\log w=\log z-\partial_{t_0} (\frac 12\partial_{t_0}+\sum\limits_{k\geqslant 1}\frac{z^{-k}}{k} \partial_{t_k})v$, where $v=v(t_0, t_1, \bar t_1, t_2, \bar t_2,...)$ is a function from the area $t_0$ of $D_+$ and the momemts $t_k$ of $D_-$. Moreover, this function satisfies the dispersionless Hirota equation for 2D Toda lattice hierarchy. Thus for an effectivisation of Riemann theorem it is sufficiently to find a representation of $v$ in the form of Taylor series $v=\sum N(i_0 | i_1,...,i_k| \bar i_1,...,\bar i_{\bar k})t_0 t_{i_1},...,t_k \bar t_{\bar i_1},...,\bar t_{\bar i_{\bar k}}$. The numbers $N(i_0 | i_1,...,i_k | \bar i_1, ..., \bar i_{\bar k})$ for $i_\alpha, \bar i_\beta\leqslant 2$ is found in [arXiv: hep-th/0005259]. In this paper we find some recurrence relations that give a possible to find all $N(i_0\bigl| i_1,...,i_k|\bar i_1,...,\bar i_{\bar k})$.