Effectivisation of a String Solution of the 2d Toda Hierarchy and the Riemann Theorem about Complex Domains

Let 0 ∈ D+ be a connected domain with analytic boundary on the complex plane C. Then according to the Riemann theorem there exists a function w(z) = 1 r z + ∑∞ j=0 pjz −j , mapping biholomorphically D− = C \D+ to the exterior of the unit disk {w ∈ C : |w| > 1}. From Wiegmann’s and Zabrodin’s rezults it follows that this function is described by the formula logw = log z − ∂t0( 1 2∂t0 + ∑ k>1 z−k k ∂tk )v, where v = v(t0, t1, t̄1, t2, t̄2, . . . ) is a function of an infinite number of harmonic moments ti 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. 2000 Math. Subj. Class. 30C, 37K.