## A Fundamental Theorem of the Theory of Quasi-conformal Mappings of Two-dimensional Regions |

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absolute values approximate quasi-conformal mapping approximate solution arbitrary arc lengths assuming the values boundary values yQ(x bounded bounded functions coefficients conformal mapping constructed function converge uniformly coordinates corresponding to system curvatures derivatives of second derivatives with respect Dirichlet problem equation 52 equation 61 estimate existence theorem F and F finitely removed points fixed number function y(x,v functions F Hence homeomorphic integral known properties Lavrent'ev let the function Let us denote linear equations linear systems mapping takes nth order obtain order to prove partial derivatives properties of linear quasi-univalent satisfies the inequality satisfying Holder conditions satisfying the conditions second order small h solution of equation straight line streamline strip 77 strip h strongly elliptic system sufficiently small system 80 system of equations tial derivatives tion twice differentiable uniformly continuous uniformly convergent unit circle unit strip valid variation virtue of lemma virtue of property x-axis