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

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

2001-03-22

arxiv.org/abs/math/0103136v2

Mathematics

Complex Variables

9 pages, AmsTex

Scientific paper

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 $N(i_0\bigl| i_1,...,i_k|\bar i_1,...,\bar i_{\bar k})$.

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