# Asymptotic representation of the solution of a mixed problem for one class of integro-differential equations containing a small parameter

**Abstract**

An approximate method is considered for the conformal mapping of arbitrary simply connected univalent regions which may be means of some elementary function be mapped on a half-plane with an aperture of arbitrary shape cut in it (fig. 1). The mapped function is sought in the form of a section of series (H or (18). Using the method of least squares the problem in the case of series (1) is reduced to the solution of system (13) the coefficients of which are easily calculated by the recurrent formulae (10) and the formulae of numerical quadratures. In the case of series (IS) the problem is reduced to the solution of an analogous system of linear equations (26). Three examples are considered in which the results obtained are compared with exact mapping functions. In § 3 the described method is extended to the case of regions with a finite number of slits.

**Citation Example:** *Fil'chakov F. P.* Asymptotic representation of the solution of a mixed problem for one class of integro-differential equations containing a small parameter // Ukr. Mat. Zh. - 1962. - **14**, № 3. - pp. 299-307.

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