On the determination of constants in the Schwarz–Christoffel integral by P. P. Kufarev's method

Schwarz–Christoffel's integral maps simple-connected polygonal domains onto the upper half-plain and is successfully used in various applications related to problems of hydrodynamics, electrodynamics, and elasticity theory. The main difficulty in the practical use of this integral is in the determination of parameters entering into it. This paper investigates P. P. Kufarev's method (1947) for the numerical determination of parameters involved in the Schwarz–Christoffel transformation. In this method, the difficult problem of determining parameters is reduced to an easier one, namely, to numerical integration of a system of ordinary differential equations. These equations describe the motion of preimages of polygon vertices.
The paper presents an analysis of Kufarev's equations and describes properties of this transformation. The problem of initial conditions and a way to solve it are also considered. Determining constants in the Schwarz–Christoffel integral by Kufarev's method has a reliable theoretical basis, and the properties of this method are corroborated by calculations.