Geodesic
Approximate conformal mapping of a~unit circle on simply connected domain
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2014), pp. 57-67.

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We suggest a method of constructing the analytic function realizing an approximate conformal mapping of the unit disk on an arbitrary simply connected domain with the given smooth parametrically defined boundary. The method is based on a new boundary parameterization. Solution of the problem is reduced to the Fredholm integral equation of the second kind. We present the examples of three ways to solve the integral equation.
Keywords: conformal mapping, integral equation, contracting mapping. 
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E. A. Shirokova. Approximate conformal mapping of a~unit circle on simply connected domain. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2014), pp. 57-67. http://geodesic.mathdoc.fr/item/IVM_2014_3_a4/

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