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Solution of the Schwarz differential equation. (English) Zbl 0911.34004

Let \(S_0\) be a circular polygon with a finite number of vertices and arbitrary angles at these vertices in the complex \(w\)-plane. It is known that an analytic function \(w=w(\zeta)\) conformally maps a half-plane \(\operatorname{Im} (\zeta)>0\) onto \(S_0,\) if and only if it is a solution of the Schwarz equation \[ \frac{w^{(3)}}{w'}-\frac 32 \biggl(\frac{w''}{w'}\biggr)^2= R(\zeta)=\sum_{k=1}^m\biggl(\frac{(1-\nu_k^2)/2}{(\zeta-a_k)^2} +\frac{c_k}{\zeta-a_k}\biggr) \tag{SE} \] satisfying a certain boundary condition, where \(c_k\), \(k=1,\dots,m\), are real accessory parameters such that \(\sum^m_{k=1}c_k=0,\) \( \sum^m_{k=1}\bigl(a_kc_k+(1- \nu_k^2)/2\bigr)=(1-\nu^2_{m+1})/2.\) The general solution to (SE) is expressed as the quotient of linearly independent solutions to an equation of the Fuchsian type \[ u''+\frac 12 R(\zeta)u=0, \] which is equivalent to \[ v''+p(\zeta)v'+q(\zeta)v=0, \quad p(\zeta)=\sum^m_{k=1}\frac{1-\nu_k}{\zeta-a_k}, \quad q(\zeta)=\frac 12 R(\zeta)+\frac 12 p'(\zeta)^2 +\frac 14 p(\zeta)^2.\tag{LE} \] The function \(w(\zeta)\) is constructed by utilizing local solutions to (LE) expanded into convergent series and connection matrices associated with them. For the fixed vertices of \(S_0,\) the possible intervals of variation of accessory parameters are given.

MSC:

34M99 Ordinary differential equations in the complex domain
34B15 Nonlinear boundary value problems for ordinary differential equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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