Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University466010.18287/2541-7525-2013-19-6-54-60ON THE UNIFORM CONVERGENCE OF THE APPROXIMATE SOLUTION OF SINGULAR INTEGRO-DIFFERENTIAL EQUATION OF THE FIRST KINDOzhegovaA.V.morenov.sv@ssau.ruHayrullinaL.E.morenov.sv@ssau.ruKazan (Volga Region) Federal University0206201319654600206201702062017Copyright © 2013, Ozhegova A., Hayrullina L.2013The article presents the study of the boundary problem for singular inte-gro-differential equation of the first kind with Cauchy kernel. The authors introduce the pair of weight spaces to prove the correctness of the stated problem. The article states the sufficient conditions for the convergence of the general direct method, the method of orthogonal polynomials, and as a result uniform estimates for errors of approximate solution.singular integro-differential equation, approximate solution, correctness of the problem