Applied Mathematics and Computation 317 (2018) 150–159 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Optimal quadrature formulas with derivatives for Cauchy type singular integrals D.M. Akhmedov, A.R. Hayotov ∗ , Kh.M. Shadimetov Institute of Mathematics, Uzbek Academy of Sciences, 29, Do‘rmon yo‘li Street, Tashkent 100125, Uzbekistan a r t i c l e i n f o MSC: MSC65D32 Keywords: Optimal quadrature formulas The extremal function Sobolev space Optimal coefficients Cauchy type singular integral a b s t r a c t In the present paper in L (m) 2 (0, 1) space the optimal quadrature formulas with derivatives are constructed for approximate calculation of the Cauchy type singular integral. Explicit formulas for the optimal coefficients are obtained. Some numerical results are presented. © 2017 Elsevier Inc. All rights reserved. 1. Introduction. Statement of the problem Many problems of science and engineering are naturally reduced to singular integral equations. Moreover plane prob- lems are reduced to one dimensional singular integral equations (see [10]). The theory of one dimensional singular integral equations is given, for example, in [7,11]. It is known that the solutions of such integral equations are expressed by sin- gular integrals. Therefore approximate calculation of singular integrals with high exactness is actual problem of numerical analysis. We consider the following quadrature formula  1 0 ϕ (x) x − t dx ∼ = n  α=0 N  β=0 C α [β ]ϕ (α) (x β ), (1.1) with the error functional ℓ(x) = ε [0,1] (x) x − t − n  α=0 N  β=0 (−1) α C α [β ]δ (α) (x − x β ) (1.2) where 0 < t < 1, C α [β ] are the coefficients, x β (∈ [0, 1]) are the nodes, N is a natural number, n = 0, m − 1, ε [0,1] (x) is the characteristic function of the interval [0, 1], δ is the Dirac delta function, ϕ is a function of the space L (m) 2 (0, 1). Here L (m) 2 (0, 1) is the Sobolev space of functions with a square integrable mth generalized derivative and equipped with the norm ‖ϕ|L (m) 2 (0, 1)‖ =   1 0 (ϕ (m) (x)) 2 dx  1/2 ∗ Corresponding author. E-mail addresses: hayotov@mail.ru, abdullo_hayotov@mail.ru (Kh.M. Shadimetov). http://dx.doi.org/10.1016/j.amc.2017.09.009 0096-3003/© 2017 Elsevier Inc. All rights reserved.